Welcome to OPFUNU’s documentation!

OPFUNU is the largest python library for cutting-edge numerical optimization benchmark functions. Contains all CEC competition functions from 2005, 2008, 2010, 2013, 2014, 2015, 2017, 2019, 2020, 2021, 2022. Besides, more than 300 traditional functions with different dimensions are implemented.

• Free software: GNU General Public License (GPL) V3 license

• Total problems: > 500 problems

• Documentation: https://opfunu.readthedocs.io

• Python versions: >= 3.7.x

• Dependencies: numpy, matplotlib

Quick Start

Installation

Install the current PyPI release:

$ pip install opfunu==1.0.3

Or install the development version from GitHub:

$ pip install git+https://github.com/thieu1995/opfunu

Install directly from source code:

$ git clone https://github.com/thieu1995/opfunu.git
$ cd opfunu
$ python setup.py install

Lib’s structure

Current Structure:

docs
examples
opfunu
    cec_based
        cec.py
        cec2005.py
        cec2008.py
        ...
        cec2021.py
        cec2022.py
    name_based
        a_func.py
        b_func.py
        ...
        y_func.py
        z_func.py
    utils
        operator.py
        visualize.py
    __init__.py
    benchmark.py
README.md
setup.py

Usage

After installation, you can import Opfunu as any other Python module:

$ python
>>> import opfunu
>>> opfunu.__version__

Let’s go through some examples.

Examples

How to get the function and use it

1st way:

from opfunu.cec_based.cec2014 import F12014

func = F12014(ndim=30)
func.evaluate(func.create_solution())

## or

from opfunu.cec_based import F102014

func = F102014(ndim=50)
func.evaluate(func.create_solution())

2nd way:

import opfunu

funcs = opfunu.get_functions_by_classname("F12014")
func = funcs[0](ndim=10)
func.evaluate(func.create_solution())

## or

all_funcs_2014 = opfunu.get_functions_based_classname("2014")
print(all_funcs_2014)

For more usage examples please look at [examples](/examples) folder.

Collaborative Libraries

In this section, we will guide you how to integrate our library into other Optimization frameworks.

Mealpy Library

For example:

from opfunu.cec_based import cec2017
f3 = cec2017.F32017(ndim=30)

from mealpy import GA, FloatVar

problem = {
    "obj_func": f3.evaluate,
    "bounds": FloatVar(lb=f3.lb, ub=f3.ub),
    "minmax": "min",
}
model = GA.BaseGA(epoch=100, pop_size=50)
gbest = model.solve(problem_dict1)
print(f"Solution: {gbest.solution}, Fit: {gbest.target.fitness}")

ScikitOpt Library

For example:

from opfunu.cec_based import cec2015
f10 = cec2015.F102015(ndim=30)

from sko.DE import DE

de = DE(func=f10.evaluate, lb=f10.lb, ub=f10.ub,
                                size_pop=50, max_iter=800)
best_x, best_y = de.run()
print(f"best_x: {best_x}, best_y: {best_y}")

Opytimizer Library

For example:

from opfunu.cec_based import cec2022
f5 = cec2022.F52022(ndim=30)

from opytimizer import Opytimizer
from opytimizer.core import Function
from opytimizer.optimizers.swarm import PSO
from opytimizer.spaces import SearchSpace

space = SearchSpace(n_agents=20, n_variables=f5.ndim,
                lower_bound=f5.lb, upper_bound=f5.ub)
optimizer = PSO()
function = Function(f5.evaluate)

opt = Opytimizer(space, optimizer, function)
opt.start(n_iterations=1000)

Function Categories

In general, unconstrained problems can be classified into two categories: test functions and real-world problems:

1. Test functions are artificial problems, and can be used to evaluate the behavior of an algorithm in sometimes diverse and difficult situations.
Artificial problems may include single global minimum, single or multiple global minima in the presence of
many local minima, long narrow valleys, null-space effects and flat surfaces. These problems can be easily manipulated and modified to test the algorithms in
diverse scenarios.

2.On the other hand, real-world problems originate from different fields such as physics, chemistry, engineering, mathematics etc. These problems are
hard to manipulate and may contain complicated algebraic or differential expressions and may require a significant amount of data to compile.

The objective functions could be characterized as:

1. continuous, discontinuous

2. linear, non-linear

3. convex, non-conxex

4. unimodal, multimodal,

5. separable and non-separable.

Before solving an optimization problem. Need to ask question:

1. What aspects of the function landscape make the optimization process difficult?
2. What type of a priori knowledge is most effective for searching particular types of function landscape?

==> To answer these questions, benchmark functions can be classified with features like modality, basins, valleys, separability and dimensionality.

1) Modality The number of ambiguous peaks in the function landscape corresponds to the modality of a function. If algorithms encounters these peaks during a search process, there is a tendency that the algorithm may be trapped in one of such peaks. This will have a negative impact on the search process, as this can direct the search away from the true optimal solutions.

2) Basins A relatively steep decline surrounding a large area is called a basin. Optimization algorithms can be easily attracted to such regions. Once in these regions, the search process of an algorithm is severely hampered. This is due to lack of information to direct the search process towards the minimum. A basin corresponds to the plateau for a maximization problem, and a problem can have multiple plateaus.

3) Valleys A valley occurs when a narrow area of little change is surrounded by regions of steep descent. As with the basins, minimizers are initially attracted to this region. The progress of a search process of an algorithm may be slowed down considerably on the floor of the valley

4) Separability The separability is a measure of difficulty of different benchmark functions In general, separable functions are relatively easy to solve, when compared with their inseperable counterpart, because each variable of a function is independent of the other variables. If all the parameters or variables are independent, then a sequence of n independent optimization processes can be performed. In other words, a function of p variables is called separable, if it can written as a sum of p functions of just one variable On the other hand, a function is called non-separable, if its variables show inter-relation among themselves or are not independent If the objective function variables are independent of each other, then the objective functions can be decomposed into sub-objective function. Then, each of these sub-objectives involves only one decision variable, while treating all the others as constant.

5) Dimensionality The difficulty of a problem generally increases with its dimensionality. When the number of parameters or dimension increases, the search space also increases exponentially. For highly nonlinear problems, this dimensionality may be a significant barrier for almost all optimization algorithms.

• Multimodal: A function with more than one local optimum. The one has many local minima are among the most difficult class of problems for many algorithms.

• Functions with flat surfaces pose a difficulty for the algorithms, since the flatness of the function does not give the algorithm any information to direct

the search process towards the minima.

opfunu.benchmark module

class opfunu.benchmark.Benchmark

Bases: object

Defines an abstract class for optimization benchmark problem.

All subclasses should implement the evaluate method for a particular optimization problem.

bounds

The lower/upper bounds of the problem. This a 2D-matrix of [lower, upper] array that contain the lower and upper bounds. By default, each problem has its own bounds. But user can try to put different bounds to test the problem.

Type

list

ndim

The dimensionality of the problem. It is calculated from bounds

Type

int

lb

The lower bounds for the problem

Type

np.ndarray

ub

The upper bounds for the problem

Type

np.ndarray

f_global

The global optimum of the evaluated function.

Type

float

x_global

A list of vectors that provide the locations of the global minimum. Note that some problems have multiple global minima, not all of which may be listed.

Type

np.ndarray

n_fe

The number of function evaluations that the object has been asked to calculate.

Type

int

dim_changeable

Whether we can change the benchmark function x variable length (i.e., the dimensionality of the problem)

Type

bool

property bounds

The lower/upper bounds to be used for optimization problem. This a 2D-matrix of [lower, upper] array that contain the lower and upper bounds for the problem. The problem should not be asked for evaluation outside these bounds. len(bounds) == ndim.

check_ndim_and_bounds(ndim=None, bounds=None, default_bounds=None)

Check the bounds when initializing the object.

Parameters

• ndim (int) – The number of dimensions (variables)

• bounds (list, tuple, np.ndarray) – List of lower bound and upper bound, should use default None value

• default_bounds (np.ndarray) – List of initial lower bound and upper bound values

check_solution(x)

Raise the error if the problem size is not equal to the solution length

Parameters

x (np.ndarray) – The solution

continuous = True

convex = True

create_solution() → numpy.ndarray

Create a random solution for the current problem

Returns

solution – The random solution

Return type

1D-vector

differentiable = True

evaluate(x)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

get_paras()

Return the parameters of the problem. Depended on function

is_ndim_compatible(ndim)

Method to support searching the functions with input ndim

Parameters

ndim (int) – The number of dimensions

Returns

val – Always true if dim_changeable = True, Else return ndim == self.ndim

Return type

bool

is_succeed(x, tol=1e-05)

Check if a candidate solution at the global minimum.

Parameters

• x (np.ndarray) – The candidate vector for testing if the global minimum has been reached. Must have len(x) == self.ndim

• tol (float) – The evaluated function and known global minimum must differ by less than this amount to be at a global minimum.

Returns

is_succeed – Answer the question: is the candidate vector at the global minimum?

Return type

bool

latex_formula = 'f(\\mathbf{x})'

latex_formula_bounds = 'x_i \\in [-2\\pi, 2\\pi], \\forall i \\in \\llbracket 1, d\\rrbracket'

latex_formula_dimension = 'd \\in \\mathbb{N}_{+}^{*}'

latex_formula_global_optimum = 'f(0, ..., 0)=-1, \\text{ for}, m=5, \\beta=15'

property lb

The lower bounds for the problem

Returns

lb – The lower bounds for the problem

Return type

1D-vector

linear = False

modality = True

name = 'Benchmark name'

property ndim

The dimensionality of the problem.

Returns

ndim – The dimensionality of the problem

Return type

int

parametric = True

randomized_term = False

scalable = True

separable = False

property ub

The upper bounds for the problem

Returns

ub – The upper bounds for the problem

Return type

1D-vector

unimodal = False

opfunu.cec_based package

opfunu.cec_based.cec module

class opfunu.cec_based.cec.CecBenchmark

Bases: opfunu.benchmark.Benchmark, abc.ABC

Defines an abstract class for optimization benchmark problem.

All subclasses should implement the evaluate method for a particular optimization problem.

bounds

The lower/upper bounds of the problem. This a 2D-matrix of [lower, upper] array that contain the lower and upper bounds. By default, each problem has its own bounds. But user can try to put different bounds to test the problem.

Type

list

ndim

The dimensionality of the problem. It is calculated from bounds

Type

int

lb

The lower bounds for the problem

Type

np.ndarray

ub

The upper bounds for the problem

Type

np.ndarray

f_global

The global optimum of the evaluated function.

Type

float

x_global

A list of vectors that provide the locations of the global minimum. Note that some problems have multiple global minima, not all of which may be listed.

Type

np.ndarray

n_fe

The number of function evaluations that the object has been asked to calculate.

Type

int

dim_changeable

Whether we can change the benchmark function x variable length (i.e., the dimensionality of the problem)

Type

bool

check_m_group(m_group=None)

check_matrix_data(f_matrix, needed_dim=True)

check_ndim_and_bounds(ndim=None, dim_max=None, bounds=None, default_bounds=None)

Check the bounds when initializing the object.

Parameters

• ndim (int) – The number of dimensions (variables)

• dim_max (int) – The maximum number of dimensions (variables) that the problem is supported

• bounds (list, tuple, np.ndarray) – List of lower bound and upper bound, should use default None value

• default_bounds (np.ndarray) – List of initial lower bound and upper bound values

check_shift_data(f_shift)

check_shift_matrix(f_shift, selected_idx=None)

check_shuffle_data(f_shuffle, needed_dim=True)

check_solution(x, dim_max=None, dim_support=None)

Raise the error if the problem size is not equal to the solution length

Parameters

• x (np.ndarray) – The solution

• dim_max (The maximum number of variables that the function is supported) –

• dim_support (List of the supported dimensions) –

continuous = True

convex = True

differentiable = True

latex_formula = 'f(\\mathbf{x})'

latex_formula_bounds = 'x_i \\in [-2\\pi, 2\\pi], \\forall i \\in \\llbracket 1, d\\rrbracket'

latex_formula_dimension = 'd \\in \\mathbb{N}_{+}^{*}'

latex_formula_global_optimum = 'f(0, ..., 0)=-1, \\text{ for}, m=5, \\beta=15'

linear = False

load_matrix_data(filename=None)

load_shift_and_matrix_data(filename=None)

load_shift_data(filename=None)

load_two_matrix_and_shift_data(filename=None)

make_support_data_path(data_name)

modality = True

name = 'Benchmark name'

parametric = True

randomized_term = False

rotated = False

scalable = True

separable = False

shifted = True

unimodal = False

opfunu.cec_based.cec2005 module

class opfunu.cec_based.cec2005.F102005(ndim=None, bounds=None, f_shift='data_rastrigin', f_matrix='rastrigin_M_D', f_bias=- 330.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Suganthan, P.N., Hansen, N., Liang, J.J., Deb, K., Chen, Y.P., Auger, A. and Tiwari, S., 2005.

Problem definitions and evaluation criteria for the CEC 2005 special session on real-parameter optimization. KanGAL report, 2005005(2005), p.2005.

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_6(x) = \\sum_{i=1}^D \\Big(100(z_i^2 - z_{i+1})^2 + (z_i-1)^2 \\Big) + bias; z=x-o+1;\\x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_6(x^*) = bias = 390.0'

linear = False

modality = True

name = 'F10: Shifted Rotated Rastrigin’s Function'

parametric = True

randomized_term = False

rotated = True

scalable = True

separable = False

shifted = True

unimodal = False

class opfunu.cec_based.cec2005.F112005(ndim=None, bounds=None, f_shift='data_weierstrass', f_matrix='weierstrass_M_D', f_bias=90.0, a=0.5, b=3, k_max=20)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Suganthan, P.N., Hansen, N., Liang, J.J., Deb, K., Chen, Y.P., Auger, A. and Tiwari, S., 2005.

Problem definitions and evaluation criteria for the CEC 2005 special session on real-parameter optimization. KanGAL report, 2005005(2005), p.2005.

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_6(x) = \\sum_{i=1}^D \\Big(100(z_i^2 - z_{i+1})^2 + (z_i-1)^2 \\Big) + bias; z=x-o+1;\\x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_6(x^*) = bias = 390.0'

linear = False

modality = True

name = 'F11: Shifted Rotated Weierstrass Function'

parametric = True

randomized_term = False

rotated = True

scalable = True

separable = False

shifted = True

unimodal = False

class opfunu.cec_based.cec2005.F12005(ndim=None, bounds=None, f_shift='data_sphere', f_bias=- 450.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Suganthan, P.N., Hansen, N., Liang, J.J., Deb, K., Chen, Y.P., Auger, A. and Tiwari, S., 2005.

Problem definitions and evaluation criteria for the CEC 2005 special session on real-parameter optimization. KanGAL report, 2005005(2005), p.2005.

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = -450.0'

linear = False

modality = True

name = 'F1: Shifted Sphere Function'

parametric = True

randomized_term = False

rotated = False

scalable = True

separable = True

shifted = True

unimodal = True

class opfunu.cec_based.cec2005.F122005(ndim=None, bounds=None, f_shift='data_schwefel_213', f_bias=- 460.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Suganthan, P.N., Hansen, N., Liang, J.J., Deb, K., Chen, Y.P., Auger, A. and Tiwari, S., 2005.

Problem definitions and evaluation criteria for the CEC 2005 special session on real-parameter optimization. KanGAL report, 2005005(2005), p.2005.

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_6(x) = \\sum_{i=1}^D \\Big(100(z_i^2 - z_{i+1})^2 + (z_i-1)^2 \\Big) + bias; z=x-o+1;\\x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_6(x^*) = bias = 390.0'

linear = False

modality = True

name = 'F12: Schwefel’s Problem 2.13'

parametric = True

randomized_term = False

rotated = False

scalable = True

separable = False

shifted = True

unimodal = False

class opfunu.cec_based.cec2005.F132005(ndim=None, bounds=None, f_shift='data_EF8F2', f_bias=- 130.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Suganthan, P.N., Hansen, N., Liang, J.J., Deb, K., Chen, Y.P., Auger, A. and Tiwari, S., 2005.

Problem definitions and evaluation criteria for the CEC 2005 special session on real-parameter optimization. KanGAL report, 2005005(2005), p.2005.

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_5(x) = max{\\Big| A_ix - B_i \\Big|} + bias; i=1,...,D; x=[x_1, ..., x_D];\\\\A: \\text{is D*D matrix}, a_{ij}: \\text{are integer random numbers in range [-500, 500]};\\\\det(A) \\neq 0; A_i: \\text{is the } i^{th} \\text{ row of A.}\\\\B_i = A_i * o, o=[o_1, ..., o_D]: \\text{the shifted global optimum}\\\\ \\text{After load the data file, set } o_i=-100, \\text{ for } i=1,2,...[D/4], \\text{and }o_i=100 \\text{ for } i=[3D/4,...,D]'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_5(x^*) = bias = -310.0'

linear = False

modality = True

name = 'F13: Shifted Expanded Griewank’s plus Rosenbrock’s Function (F8F2)'

parametric = True

randomized_term = False

rotated = False

scalable = True

separable = False

shifted = True

unimodal = False

class opfunu.cec_based.cec2005.F142005(ndim=None, bounds=None, f_shift='data_E_ScafferF6', f_matrix='E_ScafferF6_M_D', f_bias=- 300.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Suganthan, P.N., Hansen, N., Liang, J.J., Deb, K., Chen, Y.P., Auger, A. and Tiwari, S., 2005.

Problem definitions and evaluation criteria for the CEC 2005 special session on real-parameter optimization. KanGAL report, 2005005(2005), p.2005.

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_6(x) = \\sum_{i=1}^D \\Big(100(z_i^2 - z_{i+1})^2 + (z_i-1)^2 \\Big) + bias; z=x-o+1;\\x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_6(x^*) = bias = 390.0'

linear = False

modality = True

name = 'F14: Shifted Rotated Expanded Scaffer’s F6 Function'

parametric = True

randomized_term = False

rotated = False

scalable = True

separable = False

shifted = True

unimodal = False

class opfunu.cec_based.cec2005.F152005(ndim=None, bounds=None, f_shift='data_hybrid_func1', f_bias=120.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Suganthan, P.N., Hansen, N., Liang, J.J., Deb, K., Chen, Y.P., Auger, A. and Tiwari, S., 2005.

Problem definitions and evaluation criteria for the CEC 2005 special session on real-parameter optimization. KanGAL report, 2005005(2005), p.2005.

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

fi__(x, idx)

latex_formula = 'F_6(x) = \\sum_{i=1}^D \\Big(100(z_i^2 - z_{i+1})^2 + (z_i-1)^2 \\Big) + bias; z=x-o+1;\\x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_6(x^*) = bias = 390.0'

linear = False

modality = True

name = 'F15: Hybrid Composition Function'

parametric = True

randomized_term = False

rotated = False

scalable = True

separable = False

shifted = True

unimodal = False

class opfunu.cec_based.cec2005.F162005(ndim=None, bounds=None, f_shift='data_hybrid_func1', f_matrix='hybrid_func1_M_D', f_bias=120.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Suganthan, P.N., Hansen, N., Liang, J.J., Deb, K., Chen, Y.P., Auger, A. and Tiwari, S., 2005.

Problem definitions and evaluation criteria for the CEC 2005 special session on real-parameter optimization. KanGAL report, 2005005(2005), p.2005.

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

fi__(x, idx)

latex_formula = 'F_6(x) = \\sum_{i=1}^D \\Big(100(z_i^2 - z_{i+1})^2 + (z_i-1)^2 \\Big) + bias; z=x-o+1;\\x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_6(x^*) = bias = 390.0'

linear = False

modality = True

name = 'F16: Rotated Version of Hybrid Composition Function F15'

parametric = True

randomized_term = False

rotated = True

scalable = True

separable = False

shifted = True

unimodal = False

class opfunu.cec_based.cec2005.F172005(ndim=None, bounds=None, f_shift='data_hybrid_func1', f_matrix='hybrid_func1_M_D', f_bias=120.0)

Bases: opfunu.cec_based.cec2005.F162005

1

Suganthan, P.N., Hansen, N., Liang, J.J., Deb, K., Chen, Y.P., Auger, A. and Tiwari, S., 2005.

Problem definitions and evaluation criteria for the CEC 2005 special session on real-parameter optimization. KanGAL report, 2005005(2005), p.2005.

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_6(x) = \\sum_{i=1}^D \\Big(100(z_i^2 - z_{i+1})^2 + (z_i-1)^2 \\Big) + bias; z=x-o+1;\\x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_6(x^*) = bias = 390.0'

name = 'F17: F16 with Noise in Fitness'

randomized_term = True

class opfunu.cec_based.cec2005.F182005(ndim=None, bounds=None, f_shift='data_hybrid_func2', f_matrix='hybrid_func2_M_D', f_bias=10.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Suganthan, P.N., Hansen, N., Liang, J.J., Deb, K., Chen, Y.P., Auger, A. and Tiwari, S., 2005.

Problem definitions and evaluation criteria for the CEC 2005 special session on real-parameter optimization. KanGAL report, 2005005(2005), p.2005.

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

fi__(x, idx)

latex_formula = 'F_6(x) = \\sum_{i=1}^D \\Big(100(z_i^2 - z_{i+1})^2 + (z_i-1)^2 \\Big) + bias; z=x-o+1;\\x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_6(x^*) = bias = 390.0'

linear = False

modality = True

name = 'F18: Rotated Hybrid Composition Function'

parametric = True

randomized_term = False

rotated = True

scalable = True

separable = False

shifted = True

unimodal = False

class opfunu.cec_based.cec2005.F192005(ndim=None, bounds=None, f_shift='data_hybrid_func2', f_matrix='hybrid_func2_M_D', f_bias=10.0)

Bases: opfunu.cec_based.cec2005.F182005

1

Suganthan, P.N., Hansen, N., Liang, J.J., Deb, K., Chen, Y.P., Auger, A. and Tiwari, S., 2005.

Problem definitions and evaluation criteria for the CEC 2005 special session on real-parameter optimization. KanGAL report, 2005005(2005), p.2005.

latex_formula = 'F_6(x) = \\sum_{i=1}^D \\Big(100(z_i^2 - z_{i+1})^2 + (z_i-1)^2 \\Big) + bias; z=x-o+1;\\x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_6(x^*) = bias = 390.0'

name = 'F19: Rotated Hybrid Composition Function with narrow basin global optimum'

class opfunu.cec_based.cec2005.F202005(ndim=None, bounds=None, f_shift='data_hybrid_func2', f_matrix='hybrid_func2_M_D', f_bias=10.0)

Bases: opfunu.cec_based.cec2005.F182005

1

Suganthan, P.N., Hansen, N., Liang, J.J., Deb, K., Chen, Y.P., Auger, A. and Tiwari, S., 2005.

Problem definitions and evaluation criteria for the CEC 2005 special session on real-parameter optimization. KanGAL report, 2005005(2005), p.2005.

latex_formula = 'F_6(x) = \\sum_{i=1}^D \\Big(100(z_i^2 - z_{i+1})^2 + (z_i-1)^2 \\Big) + bias; z=x-o+1;\\x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_6(x^*) = bias = 390.0'

name = 'F20: Rotated Hybrid Composition Function with Global Optimum on the Bounds'

class opfunu.cec_based.cec2005.F212005(ndim=None, bounds=None, f_shift='data_hybrid_func3', f_matrix='hybrid_func3_M_D', f_bias=360.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Suganthan, P.N., Hansen, N., Liang, J.J., Deb, K., Chen, Y.P., Auger, A. and Tiwari, S., 2005.

Problem definitions and evaluation criteria for the CEC 2005 special session on real-parameter optimization. KanGAL report, 2005005(2005), p.2005.

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

fi__(x, idx)

latex_formula = 'F_6(x) = \\sum_{i=1}^D \\Big(100(z_i^2 - z_{i+1})^2 + (z_i-1)^2 \\Big) + bias; z=x-o+1;\\x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_6(x^*) = bias = 390.0'

linear = False

modality = True

name = 'F21: Rotated Hybrid Composition Function'

parametric = True

randomized_term = False

rotated = True

scalable = True

separable = False

shifted = True

unimodal = False

class opfunu.cec_based.cec2005.F22005(ndim=None, bounds=None, f_shift='data_schwefel_102', f_bias=- 450.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Suganthan, P.N., Hansen, N., Liang, J.J., Deb, K., Chen, Y.P., Auger, A. and Tiwari, S., 2005.

Problem definitions and evaluation criteria for the CEC 2005 special session on real-parameter optimization. KanGAL report, 2005005(2005), p.2005.

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_2(x) = \\sum_{i=1}^D (\\sum_{j=1}^i z_j)^2 + bias, z=x-o, \\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_2(x^*) = bias = -450.0'

linear = False

modality = True

name = 'F2: Shifted Schwefel’s Problem 1.2'

parametric = True

randomized_term = False

rotated = False

scalable = True

separable = False

shifted = True

unimodal = True

class opfunu.cec_based.cec2005.F222005(ndim=None, bounds=None, f_shift='data_hybrid_func3', f_matrix='hybrid_func3_HM_D', f_bias=360.0)

Bases: opfunu.cec_based.cec2005.F212005

1

Suganthan, P.N., Hansen, N., Liang, J.J., Deb, K., Chen, Y.P., Auger, A. and Tiwari, S., 2005.

Problem definitions and evaluation criteria for the CEC 2005 special session on real-parameter optimization. KanGAL report, 2005005(2005), p.2005.

latex_formula = 'F_6(x) = \\sum_{i=1}^D \\Big(100(z_i^2 - z_{i+1})^2 + (z_i-1)^2 \\Big) + bias; z=x-o+1;\\x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_6(x^*) = bias = 390.0'

name = 'F22: Rotated Hybrid Composition Function with High Condition Number Matrix'

class opfunu.cec_based.cec2005.F232005(ndim=None, bounds=None, f_shift='data_hybrid_func3', f_matrix='hybrid_func3_M_D', f_bias=360.0)

Bases: opfunu.cec_based.cec2005.F212005

1

Suganthan, P.N., Hansen, N., Liang, J.J., Deb, K., Chen, Y.P., Auger, A. and Tiwari, S., 2005.

Problem definitions and evaluation criteria for the CEC 2005 special session on real-parameter optimization. KanGAL report, 2005005(2005), p.2005.

continuous = False

differentiable = False

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_6(x) = \\sum_{i=1}^D \\Big(100(z_i^2 - z_{i+1})^2 + (z_i-1)^2 \\Big) + bias; z=x-o+1;\\x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_6(x^*) = bias = 390.0'

name = 'F21: Rotated Hybrid Composition Function'

class opfunu.cec_based.cec2005.F242005(ndim=None, bounds=None, f_shift='data_hybrid_func4', f_matrix='hybrid_func4_M_D', f_bias=260.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Suganthan, P.N., Hansen, N., Liang, J.J., Deb, K., Chen, Y.P., Auger, A. and Tiwari, S., 2005.

Problem definitions and evaluation criteria for the CEC 2005 special session on real-parameter optimization. KanGAL report, 2005005(2005), p.2005.

continuous = False

convex = False

differentiable = False

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

fi__(x, idx)

latex_formula = 'F_6(x) = \\sum_{i=1}^D \\Big(100(z_i^2 - z_{i+1})^2 + (z_i-1)^2 \\Big) + bias; z=x-o+1;\\x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_6(x^*) = bias = 390.0'

linear = False

modality = True

name = 'F24: Rotated Hybrid Composition Function'

parametric = True

randomized_term = True

rotated = True

scalable = True

separable = False

shifted = True

unimodal = False

class opfunu.cec_based.cec2005.F252005(ndim=None, bounds=None, f_shift='data_hybrid_func4', f_matrix='hybrid_func4_M_D', f_bias=260.0)

Bases: opfunu.cec_based.cec2005.F242005

1

Suganthan, P.N., Hansen, N., Liang, J.J., Deb, K., Chen, Y.P., Auger, A. and Tiwari, S., 2005.

Problem definitions and evaluation criteria for the CEC 2005 special session on real-parameter optimization. KanGAL report, 2005005(2005), p.2005.

latex_formula = 'F_6(x) = \\sum_{i=1}^D \\Big(100(z_i^2 - z_{i+1})^2 + (z_i-1)^2 \\Big) + bias; z=x-o+1;\\x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_6(x^*) = bias = 390.0'

name = 'F25: Rotated Hybrid Composition Function without bounds'

class opfunu.cec_based.cec2005.F32005(ndim=None, bounds=None, f_shift='data_high_cond_elliptic_rot', f_matrix='elliptic_M_D', f_bias=- 450.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Suganthan, P.N., Hansen, N., Liang, J.J., Deb, K., Chen, Y.P., Auger, A. and Tiwari, S., 2005.

Problem definitions and evaluation criteria for the CEC 2005 special session on real-parameter optimization. KanGAL report, 2005005(2005), p.2005.

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_3(x) = \\sum_{i=1}^D (10^6)^{\\frac{i-1}{D-1}} z_i^2 + bias; \\\\ z=(x-o).M; x=[x_1, ..., x_D], \\\\o=[o_1, ..., o_D]: \\text{the shifted global optimum}\\\\ M: \\text{orthogonal matrix}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in [1, D]'

latex_formula_dimension = 'D \\in [10, 30, 50]'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_3(x^*) = bias = -450.0'

linear = False

modality = True

name = 'F3: Shifted Rotated High Conditioned Elliptic Function'

parametric = True

randomized_term = False

rotated = False

scalable = True

separable = False

shifted = True

unimodal = True

class opfunu.cec_based.cec2005.F42005(ndim=None, bounds=None, f_shift='data_schwefel_102', f_bias=- 450.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Suganthan, P.N., Hansen, N., Liang, J.J., Deb, K., Chen, Y.P., Auger, A. and Tiwari, S., 2005.

Problem definitions and evaluation criteria for the CEC 2005 special session on real-parameter optimization. KanGAL report, 2005005(2005), p.2005.

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_4(x) = \\Big(\\sum_{i=1}^D (\\sum_{j=1}^i)^2\\Big)*\\Big(1 + 0.4|N(0, 1)|\\Big)+ bias;\\\\ z=(x-o).M; x=[x_1, ..., x_D], \\\\o=[o_1, ..., o_D]: \\text{the shifted global optimum}\\\\ N(0,1): \\text{gaussian noise}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_4(x^*) = bias = -450.0'

linear = False

modality = True

name = 'F4: Shifted Schwefel’s Problem 1.2 with Noise in Fitness'

parametric = True

randomized_term = True

rotated = False

scalable = True

separable = False

shifted = True

unimodal = True

class opfunu.cec_based.cec2005.F52005(ndim=None, bounds=None, f_shift='data_schwefel_206', f_bias=- 310.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Suganthan, P.N., Hansen, N., Liang, J.J., Deb, K., Chen, Y.P., Auger, A. and Tiwari, S., 2005.

Problem definitions and evaluation criteria for the CEC 2005 special session on real-parameter optimization. KanGAL report, 2005005(2005), p.2005.

continuous = False

convex = True

differentiable = False

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_5(x) = max{\\Big| A_ix - B_i \\Big|} + bias; i=1,...,D; x=[x_1, ..., x_D];\\\\A: \\text{is D*D matrix}, a_{ij}: \\text{are integer random numbers in range [-500, 500]};\\\\det(A) \\neq 0; A_i: \\text{is the } i^{th} \\text{ row of A.}\\\\B_i = A_i * o, o=[o_1, ..., o_D]: \\text{the shifted global optimum}\\\\ \\text{After load the data file, set } o_i=-100, \\text{ for } i=1,2,...[D/4], \\text{and }o_i=100 \\text{ for } i=[3D/4,...,D]'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_5(x^*) = bias = -310.0'

linear = True

modality = True

name = 'F5: Schwefel’s Problem 2.6 with Global Optimum on Bounds'

parametric = True

randomized_term = False

rotated = False

scalable = True

separable = False

shifted = True

unimodal = True

class opfunu.cec_based.cec2005.F62005(ndim=None, bounds=None, f_shift='data_rosenbrock', f_bias=390.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Suganthan, P.N., Hansen, N., Liang, J.J., Deb, K., Chen, Y.P., Auger, A. and Tiwari, S., 2005.

Problem definitions and evaluation criteria for the CEC 2005 special session on real-parameter optimization. KanGAL report, 2005005(2005), p.2005.

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_6(x) = \\sum_{i=1}^D \\Big(100(z_i^2 - z_{i+1})^2 + (z_i-1)^2 \\Big) + bias; z=x-o+1;\\x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_6(x^*) = bias = 390.0'

linear = False

modality = False

name = 'F6: Shifted Rosenbrock’s Function'

parametric = True

randomized_term = False

rotated = False

scalable = True

separable = False

shifted = True

unimodal = False

class opfunu.cec_based.cec2005.F72005(ndim=None, bounds=None, f_shift='data_griewank', f_matrix='griewank_M_D', f_bias=- 180.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Suganthan, P.N., Hansen, N., Liang, J.J., Deb, K., Chen, Y.P., Auger, A. and Tiwari, S., 2005.

Problem definitions and evaluation criteria for the CEC 2005 special session on real-parameter optimization. KanGAL report, 2005005(2005), p.2005.

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_6(x) = \\sum_{i=1}^D \\Big(100(z_i^2 - z_{i+1})^2 + (z_i-1)^2 \\Big) + bias; z=x-o+1;\\x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_6(x^*) = bias = 390.0'

linear = False

modality = False

name = 'F7: Shifted Rotated Griewank’s Function without Bounds'

parametric = True

randomized_term = True

rotated = True

scalable = True

separable = False

shifted = True

unimodal = False

class opfunu.cec_based.cec2005.F82005(ndim=None, bounds=None, f_shift='data_ackley', f_matrix='ackley_M_D', f_bias=- 140.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Suganthan, P.N., Hansen, N., Liang, J.J., Deb, K., Chen, Y.P., Auger, A. and Tiwari, S., 2005.

Problem definitions and evaluation criteria for the CEC 2005 special session on real-parameter optimization. KanGAL report, 2005005(2005), p.2005.

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_6(x) = \\sum_{i=1}^D \\Big(100(z_i^2 - z_{i+1})^2 + (z_i-1)^2 \\Big) + bias; z=x-o+1;\\x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_6(x^*) = bias = 390.0'

linear = False

modality = True

name = 'F8: Shifted Rotated Ackley’s Function with Global Optimum on Bounds'

parametric = True

randomized_term = True

rotated = True

scalable = True

separable = False

shifted = True

unimodal = False

class opfunu.cec_based.cec2005.F92005(ndim=None, bounds=None, f_shift='data_rastrigin', f_bias=- 330.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Suganthan, P.N., Hansen, N., Liang, J.J., Deb, K., Chen, Y.P., Auger, A. and Tiwari, S., 2005.

Problem definitions and evaluation criteria for the CEC 2005 special session on real-parameter optimization. KanGAL report, 2005005(2005), p.2005.

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_5(x) = max{\\Big| A_ix - B_i \\Big|} + bias; i=1,...,D; x=[x_1, ..., x_D];\\\\A: \\text{is D*D matrix}, a_{ij}: \\text{are integer random numbers in range [-500, 500]};\\\\det(A) \\neq 0; A_i: \\text{is the } i^{th} \\text{ row of A.}\\\\B_i = A_i * o, o=[o_1, ..., o_D]: \\text{the shifted global optimum}\\\\ \\text{After load the data file, set } o_i=-100, \\text{ for } i=1,2,...[D/4], \\text{and }o_i=100 \\text{ for } i=[3D/4,...,D]'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_5(x^*) = bias = -310.0'

linear = False

modality = True

name = 'F9: Shifted Rastrigin’s Function'

parametric = True

randomized_term = False

rotated = False

scalable = True

separable = False

shifted = True

unimodal = False

opfunu.cec_based.cec2008 module

class opfunu.cec_based.cec2008.F12008(ndim=None, bounds=None, f_shift='sphere_shift_func_data', f_bias=- 450.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Tang, K., Yáo, X., Suganthan, P. N., MacNish, C., Chen, Y. P., Chen, C. M., & Yang, Z. (2007). Benchmark functions

for the CEC’2008 special session and competition on large scale global optimization. Nature inspired computation and applications laboratory, USTC, China, 24, 1-18.

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = -450.0'

linear = False

modality = False

name = 'F1: Shifted Sphere Function'

parametric = True

randomized_term = False

rotated = False

scalable = True

separable = True

shifted = True

unimodal = True

class opfunu.cec_based.cec2008.F22008(ndim=None, bounds=None, f_shift='schwefel_shift_func_data', f_bias=- 450.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Tang, K., Yáo, X., Suganthan, P. N., MacNish, C., Chen, Y. P., Chen, C. M., & Yang, Z. (2007). Benchmark functions

for the CEC’2008 special session and competition on large scale global optimization. Nature inspired computation and applications laboratory, USTC, China, 24, 1-18.

continuous = False

convex = True

differentiable = False

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = -450.0'

linear = True

modality = False

name = 'F2: Schwefel’s Problem 2.21'

parametric = True

randomized_term = False

rotated = False

scalable = True

separable = False

shifted = True

unimodal = True

class opfunu.cec_based.cec2008.F32008(ndim=None, bounds=None, f_shift='rosenbrock_shift_func_data', f_bias=- 390.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Tang, K., Yáo, X., Suganthan, P. N., MacNish, C., Chen, Y. P., Chen, C. M., & Yang, Z. (2007). Benchmark functions

for the CEC’2008 special session and competition on large scale global optimization. Nature inspired computation and applications laboratory, USTC, China, 24, 1-18.

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = -450.0'

linear = False

modality = False

name = 'F3: Shifted Rosenbrock’s Function'

parametric = True

randomized_term = False

rotated = False

scalable = True

separable = False

shifted = True

unimodal = False

class opfunu.cec_based.cec2008.F42008(ndim=None, bounds=None, f_shift='rastrigin_shift_func_data', f_bias=- 330.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Tang, K., Yáo, X., Suganthan, P. N., MacNish, C., Chen, Y. P., Chen, C. M., & Yang, Z. (2007). Benchmark functions

for the CEC’2008 special session and competition on large scale global optimization. Nature inspired computation and applications laboratory, USTC, China, 24, 1-18.

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = -450.0'

linear = False

modality = True

name = 'F4: Shifted Rastrigin’s Function'

parametric = True

randomized_term = False

rotated = False

scalable = True

separable = True

shifted = True

unimodal = False

class opfunu.cec_based.cec2008.F52008(ndim=None, bounds=None, f_shift='griewank_shift_func_data', f_bias=- 180.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Tang, K., Yáo, X., Suganthan, P. N., MacNish, C., Chen, Y. P., Chen, C. M., & Yang, Z. (2007). Benchmark functions

for the CEC’2008 special session and competition on large scale global optimization. Nature inspired computation and applications laboratory, USTC, China, 24, 1-18.

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = -450.0'

linear = False

modality = False

name = 'F5: Shifted Griewank’s Function'

parametric = True

randomized_term = False

rotated = False

scalable = True

separable = False

shifted = True

unimodal = False

class opfunu.cec_based.cec2008.F62008(ndim=None, bounds=None, f_shift='ackley_shift_func_data', f_bias=- 140.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Tang, K., Yáo, X., Suganthan, P. N., MacNish, C., Chen, Y. P., Chen, C. M., & Yang, Z. (2007). Benchmark functions

for the CEC’2008 special session and competition on large scale global optimization. Nature inspired computation and applications laboratory, USTC, China, 24, 1-18.

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = -450.0'

linear = False

modality = False

name = 'F6: Shifted Ackley’s Function'

parametric = True

randomized_term = False

rotated = False

scalable = True

separable = True

shifted = True

unimodal = False

class opfunu.cec_based.cec2008.F72008(ndim=None, bounds=None, f_shift='rastrigin_shift_func_data', f_bias=0.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Tang, K., Yáo, X., Suganthan, P. N., MacNish, C., Chen, Y. P., Chen, C. M., & Yang, Z. (2007). Benchmark functions

for the CEC’2008 special session and competition on large scale global optimization. Nature inspired computation and applications laboratory, USTC, China, 24, 1-18.

continuous = True

convex = False

differentiable = False

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = unknown, F_1(x^*) = unknown'

linear = False

modality = True

name = 'F7: FastFractal “DoubleDip” Function'

parametric = True

randomized_term = True

rotated = False

scalable = True

separable = False

shifted = True

unimodal = False

opfunu.cec_based.cec2010 module

class opfunu.cec_based.cec2010.F102010(ndim=None, bounds=None, f_shift='f10_op', f_matrix='f10_m', m_group=50)

Bases: opfunu.cec_based.cec2010.F92010

1

Benchmark Functions for the CEC’2010 Special Session and Competition on Large-Scale Global Optimization

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = 0'

name = 'F10: D/2m-group Shifted and m-rotated Rastrigin’s Function'

unimodal = False

class opfunu.cec_based.cec2010.F112010(ndim=None, bounds=None, f_shift='f11_op', f_matrix='f11_m', m_group=50)

Bases: opfunu.cec_based.cec2010.F92010

1

Benchmark Functions for the CEC’2010 Special Session and Competition on Large-Scale Global Optimization

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = 0'

name = 'F11: D/2m-group Shifted and m-rotated Ackley’s Function'

unimodal = False

class opfunu.cec_based.cec2010.F12010(ndim=None, bounds=None, f_shift='f01_o')

Bases: opfunu.cec_based.cec.CecBenchmark

1

Benchmark Functions for the CEC’2010 Special Session and Competition on Large-Scale Global Optimization

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = 0'

linear = False

modality = True

name = 'F1: Shifted Elliptic Function'

parametric = True

randomized_term = False

rotated = False

scalable = True

separable = True

shifted = True

unimodal = True

class opfunu.cec_based.cec2010.F122010(ndim=None, bounds=None, f_shift='f11_op', m_group=50)

Bases: opfunu.cec_based.cec2010.F72010

1

Benchmark Functions for the CEC’2010 Special Session and Competition on Large-Scale Global Optimization

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = 0'

name = 'F12: D/2m-group Shifted m-dimensional Schwefel’s Problem 1.2'

class opfunu.cec_based.cec2010.F132010(ndim=None, bounds=None, f_shift='f13_op', m_group=50)

Bases: opfunu.cec_based.cec2010.F72010

1

Benchmark Functions for the CEC’2010 Special Session and Competition on Large-Scale Global Optimization

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = 0'

name = 'F13: D/2m-group Shifted m-dimensional Rosenbrock’s Function'

unimodal = False

class opfunu.cec_based.cec2010.F142010(ndim=None, bounds=None, f_shift='f14_op', f_matrix='f14_m', m_group=50)

Bases: opfunu.cec_based.cec2010.F92010

1

Benchmark Functions for the CEC’2010 Special Session and Competition on Large-Scale Global Optimization

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = 0'

name = 'F14: D/m-group Shifted and m-rotated Elliptic Function'

class opfunu.cec_based.cec2010.F152010(ndim=None, bounds=None, f_shift='f15_op', f_matrix='f15_m', m_group=50)

Bases: opfunu.cec_based.cec2010.F92010

1

Benchmark Functions for the CEC’2010 Special Session and Competition on Large-Scale Global Optimization

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = 0'

name = 'F15: D/m-group Shifted and m-rotated Rastrigin’s Function'

unimodal = False

class opfunu.cec_based.cec2010.F162010(ndim=None, bounds=None, f_shift='f16_op', f_matrix='f16_m', m_group=50)

Bases: opfunu.cec_based.cec2010.F92010

1

Benchmark Functions for the CEC’2010 Special Session and Competition on Large-Scale Global Optimization

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = 0'

name = 'F16: D/m-group Shifted and m-rotated Ackley’s Function'

unimodal = False

class opfunu.cec_based.cec2010.F172010(ndim=None, bounds=None, f_shift='f17_op', m_group=50)

Bases: opfunu.cec_based.cec2010.F72010

1

Benchmark Functions for the CEC’2010 Special Session and Competition on Large-Scale Global Optimization

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = 0'

name = 'F17: D/m-group Shifted m-dimensional Schwefel’s Problem 1.2'

unimodal = True

class opfunu.cec_based.cec2010.F182010(ndim=None, bounds=None, f_shift='f18_op', m_group=50)

Bases: opfunu.cec_based.cec2010.F72010

1

Benchmark Functions for the CEC’2010 Special Session and Competition on Large-Scale Global Optimization

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = 0'

name = 'F18: D/m-group Shifted m-dimensional Rosenbrock’s Function'

unimodal = False

class opfunu.cec_based.cec2010.F192010(ndim=None, bounds=None, f_shift='f19_o')

Bases: opfunu.cec_based.cec2010.F12010

1

Benchmark Functions for the CEC’2010 Special Session and Competition on Large-Scale Global Optimization

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = 0'

name = 'F19: Shifted Schwefel’s Problem 1.2'

separable = False

class opfunu.cec_based.cec2010.F202010(ndim=None, bounds=None, f_shift='f20_o')

Bases: opfunu.cec_based.cec2010.F12010

1

Benchmark Functions for the CEC’2010 Special Session and Competition on Large-Scale Global Optimization

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = 0'

name = 'F20: Shifted Rosenbrock’s Function'

separable = False

unimodal = False

class opfunu.cec_based.cec2010.F22010(ndim=None, bounds=None, f_shift='f02_o')

Bases: opfunu.cec_based.cec2010.F12010

1

Benchmark Functions for the CEC’2010 Special Session and Competition on Large-Scale Global Optimization

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = 0'

name = 'F2: Shifted Rastrigin’s Function'

unimodal = False

class opfunu.cec_based.cec2010.F32010(ndim=None, bounds=None, f_shift='f03_o')

Bases: opfunu.cec_based.cec2010.F12010

1

Benchmark Functions for the CEC’2010 Special Session and Competition on Large-Scale Global Optimization

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = 0'

name = 'F3: Shifted Ackley’s Function'

unimodal = False

class opfunu.cec_based.cec2010.F42010(ndim=None, bounds=None, f_shift='f04_op', f_matrix='f04_m', m_group=50)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Benchmark Functions for the CEC’2010 Special Session and Competition on Large-Scale Global Optimization

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = 0'

linear = False

modality = True

name = 'F4: Single-group Shifted and m-rotated Elliptic Function'

parametric = True

randomized_term = False

rotated = True

scalable = True

separable = False

shifted = True

unimodal = True

class opfunu.cec_based.cec2010.F52010(ndim=None, bounds=None, f_shift='f05_op', f_matrix='f05_m', m_group=50)

Bases: opfunu.cec_based.cec2010.F42010

1

Benchmark Functions for the CEC’2010 Special Session and Competition on Large-Scale Global Optimization

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = 0'

name = 'F5: Single-group Shifted and m-rotated Rastrigin’s Function'

unimodal = False

class opfunu.cec_based.cec2010.F62010(ndim=None, bounds=None, f_shift='f06_op', f_matrix='f06_m', m_group=50)

Bases: opfunu.cec_based.cec2010.F42010

1

Benchmark Functions for the CEC’2010 Special Session and Competition on Large-Scale Global Optimization

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = 0'

name = 'F6: Single-group Shifted and m-rotated Ackley’s Function'

unimodal = False

class opfunu.cec_based.cec2010.F72010(ndim=None, bounds=None, f_shift='f07_op', m_group=50)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Benchmark Functions for the CEC’2010 Special Session and Competition on Large-Scale Global Optimization

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = 0'

linear = False

modality = True

name = 'F7: Single-group Shifted m-dimensional Schwefel’s Problem 1.2'

parametric = True

randomized_term = False

rotated = False

scalable = True

separable = False

shifted = True

unimodal = True

class opfunu.cec_based.cec2010.F82010(ndim=None, bounds=None, f_shift='f08_op', m_group=50)

Bases: opfunu.cec_based.cec2010.F72010

1

Benchmark Functions for the CEC’2010 Special Session and Competition on Large-Scale Global Optimization

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = 0'

name = 'F8: Single-group Shifted m-dimensional Rosenbrock’s Function'

unimodal = False

class opfunu.cec_based.cec2010.F92010(ndim=None, bounds=None, f_shift='f09_op', f_matrix='f09_m', m_group=50)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Benchmark Functions for the CEC’2010 Special Session and Competition on Large-Scale Global Optimization

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = 0'

linear = False

modality = True

name = 'F9: D/2m-group Shifted and m-rotated Elliptic Function'

parametric = True

randomized_term = False

rotated = True

scalable = True

separable = False

shifted = True

unimodal = True

opfunu.cec_based.cec2013 module

class opfunu.cec_based.cec2013.F102013(ndim=None, bounds=None, f_shift='shift_data', f_matrix='M_D', f_bias=- 500.0)

Bases: opfunu.cec_based.cec2013.F22013

1

Liang, J. J., Qu, B. Y., Suganthan, P. N., & Hernández-Díaz, A. G. (2013). Problem definitions and evaluation criteria

for the CEC 2013 special session on real-parameter optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou, China and Nanyang Technological University, Singapore, Technical Report, 201212(34), 281-295..

characteristics = []

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = -500.0'

name = 'F10: Rotated Griewank’s Function'

unimodal = False

class opfunu.cec_based.cec2013.F112013(ndim=None, bounds=None, f_shift='shift_data', f_bias=- 400.0)

Bases: opfunu.cec_based.cec2013.F12013

1

Liang, J. J., Qu, B. Y., Suganthan, P. N., & Hernández-Díaz, A. G. (2013). Problem definitions and evaluation criteria

for the CEC 2013 special session on real-parameter optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou, China and Nanyang Technological University, Singapore, Technical Report, 201212(34), 281-295..

characteristics = ['Asymmetrical', 'Local optima’s number is huge']

convex = False

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = -400.0'

modality = True

name = 'F11: Rastrigin’s Function'

unimodal = False

class opfunu.cec_based.cec2013.F12013(ndim=None, bounds=None, f_shift='shift_data', f_bias=- 1400.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Liang, J. J., Qu, B. Y., Suganthan, P. N., & Hernández-Díaz, A. G. (2013). Problem definitions and evaluation criteria

for the CEC 2013 special session on real-parameter optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou, China and Nanyang Technological University, Singapore, Technical Report, 201212(34), 281-295..

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = -1400.0'

linear = False

modality = False

name = 'F1: Sphere Function'

parametric = True

randomized_term = False

rotated = False

scalable = True

separable = True

shifted = True

unimodal = True

class opfunu.cec_based.cec2013.F122013(ndim=None, bounds=None, f_shift='shift_data', f_matrix='M_D', f_bias=- 300.0)

Bases: opfunu.cec_based.cec2013.F32013

1

Liang, J. J., Qu, B. Y., Suganthan, P. N., & Hernández-Díaz, A. G. (2013). Problem definitions and evaluation criteria

for the CEC 2013 special session on real-parameter optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou, China and Nanyang Technological University, Singapore, Technical Report, 201212(34), 281-295..

characteristics = ['Asymmetrical', 'Local optima’s number is huge']

convex = False

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = -300.0'

modality = True

name = 'F12: Rotated Rastrigin’s Function'

unimodal = False

class opfunu.cec_based.cec2013.F132013(ndim=None, bounds=None, f_shift='shift_data', f_matrix='M_D', f_bias=- 200.0)

Bases: opfunu.cec_based.cec2013.F32013

1

Liang, J. J., Qu, B. Y., Suganthan, P. N., & Hernández-Díaz, A. G. (2013). Problem definitions and evaluation criteria

for the CEC 2013 special session on real-parameter optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou, China and Nanyang Technological University, Singapore, Technical Report, 201212(34), 281-295..

characteristics = ['Asymmetrical', 'Local optima’s number is huge']

continuous = False

convex = False

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = -200.0'

linear = False

modality = True

name = 'F13: Non-continuous Rotated Rastrigin’s Function'

separable = False

unimodal = False

class opfunu.cec_based.cec2013.F142013(ndim=None, bounds=None, f_shift='shift_data', f_bias=- 100.0)

Bases: opfunu.cec_based.cec2013.F12013

1

Liang, J. J., Qu, B. Y., Suganthan, P. N., & Hernández-Díaz, A. G. (2013). Problem definitions and evaluation criteria

for the CEC 2013 special session on real-parameter optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou, China and Nanyang Technological University, Singapore, Technical Report, 201212(34), 281-295..

characteristics = ['Asymmetrical', 'Local optima’s number is huge', 'Second better local optimum is far from the global optimum']

continuous = True

convex = False

differentiable = False

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = -100.0'

modality = True

name = 'F14: Schwefel’s Function'

rotated = True

separable = False

unimodal = False

class opfunu.cec_based.cec2013.F152013(ndim=None, bounds=None, f_shift='shift_data', f_matrix='M_D', f_bias=100.0)

Bases: opfunu.cec_based.cec2013.F22013

1

Liang, J. J., Qu, B. Y., Suganthan, P. N., & Hernández-Díaz, A. G. (2013). Problem definitions and evaluation criteria

for the CEC 2013 special session on real-parameter optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou, China and Nanyang Technological University, Singapore, Technical Report, 201212(34), 281-295..

characteristics = ['Asymmetrical', 'Local optima’s number is huge', 'The second better local optimum is far from the global optimum']

continuous = True

convex = False

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 100.0'

modality = True

name = 'F15: Rotated Schwefel’s Function'

unimodal = False

class opfunu.cec_based.cec2013.F162013(ndim=None, bounds=None, f_shift='shift_data', f_matrix='M_D', f_bias=200.0)

Bases: opfunu.cec_based.cec2013.F32013

1

Liang, J. J., Qu, B. Y., Suganthan, P. N., & Hernández-Díaz, A. G. (2013). Problem definitions and evaluation criteria

for the CEC 2013 special session on real-parameter optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou, China and Nanyang Technological University, Singapore, Technical Report, 201212(34), 281-295..

characteristics = ['Asymmetrical', 'Continuous everywhere yet differentiable nowhere']

convex = False

differentiable = False

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 200.0'

modality = True

name = 'F16: Rotated Katsuura Function'

unimodal = False

class opfunu.cec_based.cec2013.F172013(ndim=None, bounds=None, f_shift='shift_data', f_bias=300.0)

Bases: opfunu.cec_based.cec2013.F12013

1

Liang, J. J., Qu, B. Y., Suganthan, P. N., & Hernández-Díaz, A. G. (2013). Problem definitions and evaluation criteria

for the CEC 2013 special session on real-parameter optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou, China and Nanyang Technological University, Singapore, Technical Report, 201212(34), 281-295..

convex = False

differentiable = False

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 300.0'

modality = True

name = 'F17: Lunacek bi-Rastrigin Function'

separable = False

unimodal = False

class opfunu.cec_based.cec2013.F182013(ndim=None, bounds=None, f_shift='shift_data', f_matrix='M_D', f_bias=400.0)

Bases: opfunu.cec_based.cec2013.F32013

1

Liang, J. J., Qu, B. Y., Suganthan, P. N., & Hernández-Díaz, A. G. (2013). Problem definitions and evaluation criteria

for the CEC 2013 special session on real-parameter optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou, China and Nanyang Technological University, Singapore, Technical Report, 201212(34), 281-295..

characteristics = ['Asymmetrical', 'Continuous everywhere yet differentiable nowhere']

continuous = True

convex = False

differentiable = False

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 400.0'

modality = False

name = 'F18: Rotated Lunacek bi-Rastrigin Function'

unimodal = False

class opfunu.cec_based.cec2013.F192013(ndim=None, bounds=None, f_shift='shift_data', f_matrix='M_D', f_bias=500.0)

Bases: opfunu.cec_based.cec2013.F22013

1

Liang, J. J., Qu, B. Y., Suganthan, P. N., & Hernández-Díaz, A. G. (2013). Problem definitions and evaluation criteria

for the CEC 2013 special session on real-parameter optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou, China and Nanyang Technological University, Singapore, Technical Report, 201212(34), 281-295..

characteristics = []

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 500.0'

modality = True

name = 'F19: Rotated Expanded Griewank’s plus Rosenbrock’s Function'

unimodal = False

class opfunu.cec_based.cec2013.F202013(ndim=None, bounds=None, f_shift='shift_data', f_matrix='M_D', f_bias=600.0)

Bases: opfunu.cec_based.cec2013.F32013

1

Liang, J. J., Qu, B. Y., Suganthan, P. N., & Hernández-Díaz, A. G. (2013). Problem definitions and evaluation criteria

for the CEC 2013 special session on real-parameter optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou, China and Nanyang Technological University, Singapore, Technical Report, 201212(34), 281-295..

characteristics = ['Asymmetrical']

convex = False

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 600.0'

modality = True

name = 'F20: Rotated Expanded Scaffer’s F6 Function'

unimodal = False

class opfunu.cec_based.cec2013.F212013(ndim=None, bounds=None, f_shift='shift_data', f_matrix='M_D', f_bias=700.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Liang, J. J., Qu, B. Y., Suganthan, P. N., & Hernández-Díaz, A. G. (2013). Problem definitions and evaluation criteria

for the CEC 2013 special session on real-parameter optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou, China and Nanyang Technological University, Singapore, Technical Report, 201212(34), 281-295..

characteristics = ['Asymmetrical', 'Different properties around different local optima']

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 700.0'

linear = False

modality = True

name = 'F21: Composition Function 1'

parametric = True

randomized_term = False

rotated = True

scalable = True

separable = False

shifted = True

unimodal = False

class opfunu.cec_based.cec2013.F22013(ndim=None, bounds=None, f_shift='shift_data', f_matrix='M_D', f_bias=- 1300.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Liang, J. J., Qu, B. Y., Suganthan, P. N., & Hernández-Díaz, A. G. (2013). Problem definitions and evaluation criteria

for the CEC 2013 special session on real-parameter optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou, China and Nanyang Technological University, Singapore, Technical Report, 201212(34), 281-295..

characteristics = ['Quadratic ill-conditioned', 'Smooth local irregularities']

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = -1300.0'

linear = False

modality = False

name = 'F2: Rotated High Conditioned Elliptic Function'

parametric = True

randomized_term = False

rotated = True

scalable = True

separable = False

shifted = True

unimodal = True

class opfunu.cec_based.cec2013.F222013(ndim=None, bounds=None, f_shift='shift_data', f_bias=800.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Liang, J. J., Qu, B. Y., Suganthan, P. N., & Hernández-Díaz, A. G. (2013). Problem definitions and evaluation criteria

for the CEC 2013 special session on real-parameter optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou, China and Nanyang Technological University, Singapore, Technical Report, 201212(34), 281-295..

characteristics = ['Asymmetrical', 'Different properties around different local optima']

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 800.0'

linear = False

modality = True

name = 'F22: Composition Function 2'

parametric = True

randomized_term = False

rotated = True

scalable = True

separable = True

shifted = True

unimodal = False

class opfunu.cec_based.cec2013.F232013(ndim=None, bounds=None, f_shift='shift_data', f_matrix='M_D', f_bias=900.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Liang, J. J., Qu, B. Y., Suganthan, P. N., & Hernández-Díaz, A. G. (2013). Problem definitions and evaluation criteria

for the CEC 2013 special session on real-parameter optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou, China and Nanyang Technological University, Singapore, Technical Report, 201212(34), 281-295..

characteristics = ['Asymmetrical', 'Different properties around different local optima']

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 900.0'

linear = False

modality = True

name = 'F23: Composition Function 3'

parametric = True

randomized_term = False

rotated = True

scalable = True

separable = False

shifted = True

unimodal = False

class opfunu.cec_based.cec2013.F242013(ndim=None, bounds=None, f_shift='shift_data', f_matrix='M_D', f_bias=1000.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Liang, J. J., Qu, B. Y., Suganthan, P. N., & Hernández-Díaz, A. G. (2013). Problem definitions and evaluation criteria

for the CEC 2013 special session on real-parameter optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou, China and Nanyang Technological University, Singapore, Technical Report, 201212(34), 281-295..

characteristics = ['Asymmetrical', 'Different properties around different local optima']

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 1000.0'

linear = False

modality = True

name = 'F24: Composition Function 4'

parametric = True

randomized_term = False

rotated = True

scalable = True

separable = False

shifted = True

unimodal = False

class opfunu.cec_based.cec2013.F252013(ndim=None, bounds=None, f_shift='shift_data', f_matrix='M_D', f_bias=1100.0)

Bases: opfunu.cec_based.cec2013.F242013

1

Liang, J. J., Qu, B. Y., Suganthan, P. N., & Hernández-Díaz, A. G. (2013). Problem definitions and evaluation criteria

for the CEC 2013 special session on real-parameter optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou, China and Nanyang Technological University, Singapore, Technical Report, 201212(34), 281-295..

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 1100.0'

name = 'F25: Composition Function 5'

class opfunu.cec_based.cec2013.F262013(ndim=None, bounds=None, f_shift='shift_data', f_matrix='M_D', f_bias=1200.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Liang, J. J., Qu, B. Y., Suganthan, P. N., & Hernández-Díaz, A. G. (2013). Problem definitions and evaluation criteria

for the CEC 2013 special session on real-parameter optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou, China and Nanyang Technological University, Singapore, Technical Report, 201212(34), 281-295..

characteristics = ['Asymmetrical', 'Different properties around different local optima']

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 1200.0'

linear = False

modality = True

name = 'F26: Composition Function 6'

parametric = True

randomized_term = False

rotated = True

scalable = True

separable = False

shifted = True

unimodal = False

class opfunu.cec_based.cec2013.F272013(ndim=None, bounds=None, f_shift='shift_data', f_matrix='M_D', f_bias=1300.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Liang, J. J., Qu, B. Y., Suganthan, P. N., & Hernández-Díaz, A. G. (2013). Problem definitions and evaluation criteria

for the CEC 2013 special session on real-parameter optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou, China and Nanyang Technological University, Singapore, Technical Report, 201212(34), 281-295..

characteristics = ['Asymmetrical', 'Different properties around different local optima']

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 1300.0'

linear = False

modality = True

name = 'F27: Composition Function 7'

parametric = True

randomized_term = False

rotated = True

scalable = True

separable = False

shifted = True

unimodal = False

class opfunu.cec_based.cec2013.F282013(ndim=None, bounds=None, f_shift='shift_data', f_matrix='M_D', f_bias=1400.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Liang, J. J., Qu, B. Y., Suganthan, P. N., & Hernández-Díaz, A. G. (2013). Problem definitions and evaluation criteria

for the CEC 2013 special session on real-parameter optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou, China and Nanyang Technological University, Singapore, Technical Report, 201212(34), 281-295..

characteristics = ['Asymmetrical', 'Different properties around different local optima']

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 1400.0'

linear = False

modality = True

name = 'F28: Composition Function 8'

parametric = True

randomized_term = False

rotated = True

scalable = True

separable = False

shifted = True

unimodal = False

class opfunu.cec_based.cec2013.F32013(ndim=None, bounds=None, f_shift='shift_data', f_matrix='M_D', f_bias=- 1200.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Liang, J. J., Qu, B. Y., Suganthan, P. N., & Hernández-Díaz, A. G. (2013). Problem definitions and evaluation criteria

for the CEC 2013 special session on real-parameter optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou, China and Nanyang Technological University, Singapore, Technical Report, 201212(34), 281-295..

characteristics = ['Smooth but narrow ridge']

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = -1200.0'

linear = False

modality = False

name = 'F3: Rotated Bent Cigar Function'

parametric = True

randomized_term = False

rotated = True

scalable = True

separable = False

shifted = True

unimodal = True

class opfunu.cec_based.cec2013.F42013(ndim=None, bounds=None, f_shift='shift_data', f_matrix='M_D', f_bias=- 1100.0)

Bases: opfunu.cec_based.cec2013.F22013

1

Liang, J. J., Qu, B. Y., Suganthan, P. N., & Hernández-Díaz, A. G. (2013). Problem definitions and evaluation criteria

for the CEC 2013 special session on real-parameter optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou, China and Nanyang Technological University, Singapore, Technical Report, 201212(34), 281-295..

characteristics = ['Asymmetrical', 'Smooth local irregularities', 'With one sensitive direction']

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = -1100.0'

name = 'F4: Rotated Discus Function'

class opfunu.cec_based.cec2013.F52013(ndim=None, bounds=None, f_shift='shift_data', f_bias=- 1000.0)

Bases: opfunu.cec_based.cec2013.F12013

1

Liang, J. J., Qu, B. Y., Suganthan, P. N., & Hernández-Díaz, A. G. (2013). Problem definitions and evaluation criteria

for the CEC 2013 special session on real-parameter optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou, China and Nanyang Technological University, Singapore, Technical Report, 201212(34), 281-295..

characteristics = ['Sensitivities of the zi-variables are different']

continuous = False

differentiable = False

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = -1000.0'

name = 'F5: Different Powers Function'

class opfunu.cec_based.cec2013.F62013(ndim=None, bounds=None, f_shift='shift_data', f_matrix='M_D', f_bias=- 900.0)

Bases: opfunu.cec_based.cec2013.F22013

1

Liang, J. J., Qu, B. Y., Suganthan, P. N., & Hernández-Díaz, A. G. (2013). Problem definitions and evaluation criteria

for the CEC 2013 special session on real-parameter optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou, China and Nanyang Technological University, Singapore, Technical Report, 201212(34), 281-295..

characteristics = ['Having a very narrow valley from local optimum to global optimum']

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = -900.0'

name = 'F6: Rotated Rosenbrock’s Function'

unimodal = False

class opfunu.cec_based.cec2013.F72013(ndim=None, bounds=None, f_shift='shift_data', f_matrix='M_D', f_bias=- 800.0)

Bases: opfunu.cec_based.cec2013.F32013

1

Liang, J. J., Qu, B. Y., Suganthan, P. N., & Hernández-Díaz, A. G. (2013). Problem definitions and evaluation criteria

for the CEC 2013 special session on real-parameter optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou, China and Nanyang Technological University, Singapore, Technical Report, 201212(34), 281-295..

characteristics = ['Asymmetrical', 'Local optima’s number is huge']

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = -800.0'

modality = True

name = 'F7: Rotated Schaffers F7 Function'

unimodal = False

class opfunu.cec_based.cec2013.F82013(ndim=None, bounds=None, f_shift='shift_data', f_matrix='M_D', f_bias=- 700.0)

Bases: opfunu.cec_based.cec2013.F32013

1

Liang, J. J., Qu, B. Y., Suganthan, P. N., & Hernández-Díaz, A. G. (2013). Problem definitions and evaluation criteria

for the CEC 2013 special session on real-parameter optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou, China and Nanyang Technological University, Singapore, Technical Report, 201212(34), 281-295..

characteristics = ['Asymmetrical']

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = -700.0'

name = 'F8: Rotated Ackley’s Function'

unimodal = False

class opfunu.cec_based.cec2013.F92013(ndim=None, bounds=None, f_shift='shift_data', f_matrix='M_D', f_bias=- 600.0, a=0.5, b=3.0, k_max=20)

Bases: opfunu.cec_based.cec2013.F32013

1

Liang, J. J., Qu, B. Y., Suganthan, P. N., & Hernández-Díaz, A. G. (2013). Problem definitions and evaluation criteria

for the CEC 2013 special session on real-parameter optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou, China and Nanyang Technological University, Singapore, Technical Report, 201212(34), 281-295..

characteristics = ['Asymmetrical', 'Continuous but differentiable only on a set of points']

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = -600.0'

modality = True

name = 'F9: Rotated Weierstrass Function'

unimodal = False

opfunu.cec_based.cec2014 module

class opfunu.cec_based.cec2014.F102014(ndim=None, bounds=None, f_shift='shift_data_10', f_bias=1000.0)

Bases: opfunu.cec_based.cec2014.F82014

1

Liang, J. J., Qu, B. Y., & Suganthan, P. N. (2013). Problem definitions and evaluation criteria for the CEC 2014

special session and competition on single objective real-parameter numerical optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou China and Technical Report, Nanyang Technological University, Singapore, 635, 490.

characteristics = ['Local optima’s number is huge', 'The second better local optimum is far from the global optimum']

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 1000.0'

name = 'F10: Shifted Schwefel’s Function'

class opfunu.cec_based.cec2014.F112014(ndim=None, bounds=None, f_shift='shift_data_11', f_matrix='M_11_D', f_bias=1100.0)

Bases: opfunu.cec_based.cec2014.F12014

1

Liang, J. J., Qu, B. Y., & Suganthan, P. N. (2013). Problem definitions and evaluation criteria for the CEC 2014

special session and competition on single objective real-parameter numerical optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou China and Technical Report, Nanyang Technological University, Singapore, 635, 490.

characteristics = ['Local optima’s number is huge', 'The second better local optimum is far from the global optimum']

convex = False

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 1100.0'

modality = True

name = 'F11: Shifted and Rotated Schwefel’s Function'

unimodal = False

class opfunu.cec_based.cec2014.F12014(ndim=None, bounds=None, f_shift='shift_data_1', f_matrix='M_1_D', f_bias=100.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Liang, J. J., Qu, B. Y., & Suganthan, P. N. (2013). Problem definitions and evaluation criteria for the CEC 2014

special session and competition on single objective real-parameter numerical optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou China and Technical Report, Nanyang Technological University, Singapore, 635, 490.

characteristics = ['Quadratic ill-conditioned', 'Smooth local irregularities']

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 100.0'

linear = False

modality = False

name = 'F1: Rotated High Conditioned Elliptic Function'

parametric = True

randomized_term = False

rotated = True

scalable = True

separable = False

shifted = True

unimodal = True

class opfunu.cec_based.cec2014.F122014(ndim=None, bounds=None, f_shift='shift_data_12', f_matrix='M_12_D', f_bias=1200.0)

Bases: opfunu.cec_based.cec2014.F12014

1

Liang, J. J., Qu, B. Y., & Suganthan, P. N. (2013). Problem definitions and evaluation criteria for the CEC 2014

special session and competition on single objective real-parameter numerical optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou China and Technical Report, Nanyang Technological University, Singapore, 635, 490.

characteristics = ['Continuous everywhere yet differentiable nowhere']

convex = False

differentiable = False

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 1200.0'

modality = True

name = 'F12: Shifted and Rotated Katsuura Function'

separable = False

unimodal = False

class opfunu.cec_based.cec2014.F132014(ndim=None, bounds=None, f_shift='shift_data_13', f_matrix='M_13_D', f_bias=1300.0)

Bases: opfunu.cec_based.cec2014.F12014

1

Liang, J. J., Qu, B. Y., & Suganthan, P. N. (2013). Problem definitions and evaluation criteria for the CEC 2014

special session and competition on single objective real-parameter numerical optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou China and Technical Report, Nanyang Technological University, Singapore, 635, 490.

characteristics = []

continuous = False

convex = True

differentiable = False

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 1300.0'

name = 'F13: Shifted and Rotated HappyCat Function'

separable = False

unimodal = False

class opfunu.cec_based.cec2014.F142014(ndim=None, bounds=None, f_shift='shift_data_14', f_matrix='M_14_D', f_bias=1400.0)

Bases: opfunu.cec_based.cec2014.F12014

1

Liang, J. J., Qu, B. Y., & Suganthan, P. N. (2013). Problem definitions and evaluation criteria for the CEC 2014

special session and competition on single objective real-parameter numerical optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou China and Technical Report, Nanyang Technological University, Singapore, 635, 490.

characteristics = []

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 1400.0'

modality = False

name = 'F14: Shifted and Rotated HGBat Function'

separable = False

unimodal = False

class opfunu.cec_based.cec2014.F152014(ndim=None, bounds=None, f_shift='shift_data_15', f_matrix='M_15_D', f_bias=1500.0)

Bases: opfunu.cec_based.cec2014.F12014

1

Liang, J. J., Qu, B. Y., & Suganthan, P. N. (2013). Problem definitions and evaluation criteria for the CEC 2014

special session and competition on single objective real-parameter numerical optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou China and Technical Report, Nanyang Technological University, Singapore, 635, 490.

characteristics = []

continuous = True

convex = False

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 1500.0'

linear = False

name = 'F15: Shifted and Rotated Expanded Griewank’s plus Rosenbrock’s Function'

unimodal = False

class opfunu.cec_based.cec2014.F162014(ndim=None, bounds=None, f_shift='shift_data_16', f_matrix='M_16_D', f_bias=1600.0)

Bases: opfunu.cec_based.cec2014.F12014

1

Liang, J. J., Qu, B. Y., & Suganthan, P. N. (2013). Problem definitions and evaluation criteria for the CEC 2014

special session and competition on single objective real-parameter numerical optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou China and Technical Report, Nanyang Technological University, Singapore, 635, 490.

characteristics = []

continuous = True

convex = False

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 1600.0'

modality = True

name = 'F16: Shifted and Rotated Expanded Scaffer’s F6 Function'

separable = False

unimodal = False

class opfunu.cec_based.cec2014.F172014(ndim=None, bounds=None, f_shift='shift_data_17', f_matrix='M_17_D', f_shuffle='shuffle_data_17_D', f_bias=1700.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Liang, J. J., Qu, B. Y., & Suganthan, P. N. (2013). Problem definitions and evaluation criteria for the CEC 2014

special session and competition on single objective real-parameter numerical optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou China and Technical Report, Nanyang Technological University, Singapore, 635, 490.

characteristics = ['Different properties for different variables subcomponents']

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 1700.0'

linear = False

modality = True

name = 'F17: Hybrid Function 1'

parametric = True

randomized_term = False

rotated = True

scalable = True

separable = False

shifted = True

unimodal = False

class opfunu.cec_based.cec2014.F182014(ndim=None, bounds=None, f_shift='shift_data_18', f_matrix='M_18_D', f_shuffle='shuffle_data_18_D', f_bias=1800.0)

Bases: opfunu.cec_based.cec2014.F172014

1

Liang, J. J., Qu, B. Y., & Suganthan, P. N. (2013). Problem definitions and evaluation criteria for the CEC 2014

special session and competition on single objective real-parameter numerical optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou China and Technical Report, Nanyang Technological University, Singapore, 635, 490.

characteristics = ['Different properties for different variables subcomponents']

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 1800.0'

linear = False

modality = True

name = 'F18: Hybrid Function 2'

parametric = True

randomized_term = False

rotated = True

scalable = True

separable = False

shifted = True

unimodal = False

class opfunu.cec_based.cec2014.F192014(ndim=None, bounds=None, f_shift='shift_data_19', f_matrix='M_19_D', f_shuffle='shuffle_data_19_D', f_bias=1900.0)

Bases: opfunu.cec_based.cec2014.F172014

1

Liang, J. J., Qu, B. Y., & Suganthan, P. N. (2013). Problem definitions and evaluation criteria for the CEC 2014

special session and competition on single objective real-parameter numerical optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou China and Technical Report, Nanyang Technological University, Singapore, 635, 490.

characteristics = ['Different properties for different variables subcomponents']

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 1900.0'

linear = False

modality = True

name = 'F19: Hybrid Function 3'

parametric = True

randomized_term = False

rotated = True

scalable = True

separable = False

shifted = True

unimodal = False

class opfunu.cec_based.cec2014.F202014(ndim=None, bounds=None, f_shift='shift_data_20', f_matrix='M_20_D', f_shuffle='shuffle_data_20_D', f_bias=2000.0)

Bases: opfunu.cec_based.cec2014.F192014

1

Liang, J. J., Qu, B. Y., & Suganthan, P. N. (2013). Problem definitions and evaluation criteria for the CEC 2014

special session and competition on single objective real-parameter numerical optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou China and Technical Report, Nanyang Technological University, Singapore, 635, 490.

characteristics = ['Different properties for different variables subcomponents']

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 2000.0'

linear = False

modality = True

name = 'F20: Hybrid Function 4'

parametric = True

randomized_term = False

rotated = True

scalable = True

separable = False

shifted = True

unimodal = False

class opfunu.cec_based.cec2014.F212014(ndim=None, bounds=None, f_shift='shift_data_21', f_matrix='M_21_D', f_shuffle='shuffle_data_21_D', f_bias=2100.0)

Bases: opfunu.cec_based.cec2014.F172014

1

Liang, J. J., Qu, B. Y., & Suganthan, P. N. (2013). Problem definitions and evaluation criteria for the CEC 2014

special session and competition on single objective real-parameter numerical optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou China and Technical Report, Nanyang Technological University, Singapore, 635, 490.

characteristics = ['Different properties for different variables subcomponents']

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 2100.0'

linear = False

modality = True

name = 'F21: Hybrid Function 5'

parametric = True

randomized_term = False

rotated = True

scalable = True

separable = False

shifted = True

unimodal = False

class opfunu.cec_based.cec2014.F22014(ndim=None, bounds=None, f_shift='shift_data_2', f_matrix='M_2_D', f_bias=200.0)

Bases: opfunu.cec_based.cec2014.F12014

1

Liang, J. J., Qu, B. Y., & Suganthan, P. N. (2013). Problem definitions and evaluation criteria for the CEC 2014

special session and competition on single objective real-parameter numerical optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou China and Technical Report, Nanyang Technological University, Singapore, 635, 490.

characteristics = ['Smooth but narrow ridge']

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 200.0'

name = 'F2: Rotated Bent Cigar Function'

class opfunu.cec_based.cec2014.F222014(ndim=None, bounds=None, f_shift='shift_data_22', f_matrix='M_22_D', f_shuffle='shuffle_data_22_D', f_bias=2200.0)

Bases: opfunu.cec_based.cec2014.F212014

1

Liang, J. J., Qu, B. Y., & Suganthan, P. N. (2013). Problem definitions and evaluation criteria for the CEC 2014

special session and competition on single objective real-parameter numerical optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou China and Technical Report, Nanyang Technological University, Singapore, 635, 490.

characteristics = ['Different properties for different variables subcomponents']

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 2200.0'

linear = False

modality = True

name = 'F22: Hybrid Function 6'

parametric = True

randomized_term = False

rotated = True

scalable = True

separable = False

shifted = True

unimodal = False

class opfunu.cec_based.cec2014.F232014(ndim=None, bounds=None, f_shift='shift_data_23', f_matrix='M_23_D', f_bias=2300.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Liang, J. J., Qu, B. Y., & Suganthan, P. N. (2013). Problem definitions and evaluation criteria for the CEC 2014

special session and competition on single objective real-parameter numerical optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou China and Technical Report, Nanyang Technological University, Singapore, 635, 490.

characteristics = ['Asymmetrical', 'Different properties around different local optima']

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 2300.0'

linear = False

modality = False

name = 'F23: Composition Function 1'

parametric = True

randomized_term = False

rotated = True

scalable = True

separable = False

shifted = True

unimodal = False

class opfunu.cec_based.cec2014.F242014(ndim=None, bounds=None, f_shift='shift_data_24', f_matrix='M_24_D', f_bias=2400.0)

Bases: opfunu.cec_based.cec2014.F232014

1

Liang, J. J., Qu, B. Y., & Suganthan, P. N. (2013). Problem definitions and evaluation criteria for the CEC 2014

special session and competition on single objective real-parameter numerical optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou China and Technical Report, Nanyang Technological University, Singapore, 635, 490.

characteristics = ['Asymmetrical', 'Different properties around different local optima']

convex = False

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 2400.0'

modality = True

name = 'F24: Composition Function 2'

class opfunu.cec_based.cec2014.F252014(ndim=None, bounds=None, f_shift='shift_data_25', f_matrix='M_25_D', f_bias=2500.0)

Bases: opfunu.cec_based.cec2014.F232014

1

Liang, J. J., Qu, B. Y., & Suganthan, P. N. (2013). Problem definitions and evaluation criteria for the CEC 2014

special session and competition on single objective real-parameter numerical optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou China and Technical Report, Nanyang Technological University, Singapore, 635, 490.

characteristics = ['Asymmetrical', 'Different properties around different local optima']

convex = False

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 2500.0'

name = 'F25: Composition Function 3'

class opfunu.cec_based.cec2014.F262014(ndim=None, bounds=None, f_shift='shift_data_26', f_matrix='M_26_D', f_bias=2600.0)

Bases: opfunu.cec_based.cec2014.F232014

1

Liang, J. J., Qu, B. Y., & Suganthan, P. N. (2013). Problem definitions and evaluation criteria for the CEC 2014

special session and competition on single objective real-parameter numerical optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou China and Technical Report, Nanyang Technological University, Singapore, 635, 490.

characteristics = ['Asymmetrical', 'Different properties around different local optima']

convex = False

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 2600.0'

modality = True

name = 'F26: Composition Function 4'

class opfunu.cec_based.cec2014.F272014(ndim=None, bounds=None, f_shift='shift_data_27', f_matrix='M_27_D', f_bias=2700.0)

Bases: opfunu.cec_based.cec2014.F232014

1

Liang, J. J., Qu, B. Y., & Suganthan, P. N. (2013). Problem definitions and evaluation criteria for the CEC 2014

special session and competition on single objective real-parameter numerical optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou China and Technical Report, Nanyang Technological University, Singapore, 635, 490.

characteristics = ['Asymmetrical', 'Different properties around different local optima']

convex = False

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 2700.0'

name = 'F27: Composition Function 5'

class opfunu.cec_based.cec2014.F282014(ndim=None, bounds=None, f_shift='shift_data_28', f_matrix='M_28_D', f_bias=2800.0)

Bases: opfunu.cec_based.cec2014.F232014

1

Liang, J. J., Qu, B. Y., & Suganthan, P. N. (2013). Problem definitions and evaluation criteria for the CEC 2014

special session and competition on single objective real-parameter numerical optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou China and Technical Report, Nanyang Technological University, Singapore, 635, 490.

characteristics = ['Asymmetrical', 'Different properties around different local optima']

convex = False

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 2800.0'

name = 'F28: Composition Function 6'

class opfunu.cec_based.cec2014.F292014(ndim=None, bounds=None, f_shift='shift_data_29', f_matrix='M_29_D', f_shuffle='shuffle_data_29_D', f_bias=2900.0)

Bases: opfunu.cec_based.cec2014.F232014

1

Liang, J. J., Qu, B. Y., & Suganthan, P. N. (2013). Problem definitions and evaluation criteria for the CEC 2014

special session and competition on single objective real-parameter numerical optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou China and Technical Report, Nanyang Technological University, Singapore, 635, 490.

characteristics = ['Asymmetrical', 'Different properties around different local optima']

convex = False

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 2900.0'

name = 'F29: Composition Function 7'

class opfunu.cec_based.cec2014.F302014(ndim=None, bounds=None, f_shift='shift_data_30', f_matrix='M_30_D', f_shuffle='shuffle_data_30_D', f_bias=3000.0)

Bases: opfunu.cec_based.cec2014.F232014

1

Liang, J. J., Qu, B. Y., & Suganthan, P. N. (2013). Problem definitions and evaluation criteria for the CEC 2014

special session and competition on single objective real-parameter numerical optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou China and Technical Report, Nanyang Technological University, Singapore, 635, 490.

characteristics = ['Asymmetrical', 'Different properties around different local optima', 'Different properties for different variables subcomponents']

convex = False

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 3000.0'

name = 'F30: Composition Function 8'

class opfunu.cec_based.cec2014.F32014(ndim=None, bounds=None, f_shift='shift_data_3', f_matrix='M_3_D', f_bias=300.0)

Bases: opfunu.cec_based.cec2014.F12014

1

Liang, J. J., Qu, B. Y., & Suganthan, P. N. (2013). Problem definitions and evaluation criteria for the CEC 2014

special session and competition on single objective real-parameter numerical optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou China and Technical Report, Nanyang Technological University, Singapore, 635, 490.

characteristics = ['With one sensitive direction']

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 300.0'

name = 'F3: Rotated Discus Function'

class opfunu.cec_based.cec2014.F42014(ndim=None, bounds=None, f_shift='shift_data_4', f_matrix='M_4_D', f_bias=400.0)

Bases: opfunu.cec_based.cec2014.F12014

1

Liang, J. J., Qu, B. Y., & Suganthan, P. N. (2013). Problem definitions and evaluation criteria for the CEC 2014

special session and competition on single objective real-parameter numerical optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou China and Technical Report, Nanyang Technological University, Singapore, 635, 490.

characteristics = ['Having a very narrow valley from local optimum to global optimum']

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 400.0'

name = 'F4: Shifted and Rotated Rosenbrock’s Function'

unimodal = False

class opfunu.cec_based.cec2014.F52014(ndim=None, bounds=None, f_shift='shift_data_5', f_matrix='M_5_D', f_bias=500.0)

Bases: opfunu.cec_based.cec2014.F12014

1

Liang, J. J., Qu, B. Y., & Suganthan, P. N. (2013). Problem definitions and evaluation criteria for the CEC 2014

special session and competition on single objective real-parameter numerical optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou China and Technical Report, Nanyang Technological University, Singapore, 635, 490.

characteristics = ['Having a very narrow valley from local optimum to global optimum']

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 500.0'

name = 'F5: Shifted and Rotated Ackley’s Function'

unimodal = False

class opfunu.cec_based.cec2014.F62014(ndim=None, bounds=None, f_shift='shift_data_6', f_matrix='M_6_D', f_bias=600.0)

Bases: opfunu.cec_based.cec2014.F12014

1

Liang, J. J., Qu, B. Y., & Suganthan, P. N. (2013). Problem definitions and evaluation criteria for the CEC 2014

special session and competition on single objective real-parameter numerical optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou China and Technical Report, Nanyang Technological University, Singapore, 635, 490.

characteristics = ['Continuous but differentiable only on a set of points']

convex = False

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 600.0'

modality = True

name = 'F6: Shifted and Rotated Weierstrass Function'

unimodal = False

class opfunu.cec_based.cec2014.F72014(ndim=None, bounds=None, f_shift='shift_data_7', f_matrix='M_7_D', f_bias=700.0)

Bases: opfunu.cec_based.cec2014.F12014

1

Liang, J. J., Qu, B. Y., & Suganthan, P. N. (2013). Problem definitions and evaluation criteria for the CEC 2014

special session and competition on single objective real-parameter numerical optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou China and Technical Report, Nanyang Technological University, Singapore, 635, 490.

characteristics = ['Continuous but differentiable only on a set of points']

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 700.0'

name = 'F7: Shifted and Rotated Griewank’s Function'

unimodal = False

class opfunu.cec_based.cec2014.F82014(ndim=None, bounds=None, f_shift='shift_data_8', f_bias=800.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Liang, J. J., Qu, B. Y., & Suganthan, P. N. (2013). Problem definitions and evaluation criteria for the CEC 2014

special session and competition on single objective real-parameter numerical optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou China and Technical Report, Nanyang Technological University, Singapore, 635, 490.

characteristics = ['Local optima’s number is huge']

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 800.0'

linear = False

modality = True

name = 'F8: Shifted Rastrigin’s Function'

parametric = True

randomized_term = False

rotated = False

scalable = True

separable = True

shifted = True

unimodal = False

class opfunu.cec_based.cec2014.F92014(ndim=None, bounds=None, f_shift='shift_data_9', f_matrix='M_9_D', f_bias=900.0)

Bases: opfunu.cec_based.cec2014.F12014

1

Liang, J. J., Qu, B. Y., & Suganthan, P. N. (2013). Problem definitions and evaluation criteria for the CEC 2014

special session and competition on single objective real-parameter numerical optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou China and Technical Report, Nanyang Technological University, Singapore, 635, 490.

characteristics = ['Local optima’s number is huge']

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 900.0'

modality = True

name = 'F9: Shifted and Rotated Rastrigin’s Function'

parametric = True

rotated = True

separable = False

shifted = True

unimodal = False

opfunu.cec_based.cec2015 module

class opfunu.cec_based.cec2015.F102015(ndim=None, bounds=None, f_shift='shift_data_10_D', f_matrix='M_10_D', f_shuffle='shuffle_data_10_D', f_bias=1000.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Liang, J. J., Qu, B. Y., & Suganthan, P. N. (2013). Problem definitions and evaluation criteria for the CEC 2014

special session and competition on single objective real-parameter numerical optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou China and Technical Report, Nanyang Technological University, Singapore, 635, 490.

characteristics = []

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 1000.0'

linear = False

modality = True

name = 'F10: Hybrid Function 1 (N=3)'

parametric = True

randomized_term = False

rotated = True

scalable = True

separable = False

shifted = True

unimodal = False

class opfunu.cec_based.cec2015.F112015(ndim=None, bounds=None, f_shift='shift_data_11_D', f_matrix='M_11_D', f_shuffle='shuffle_data_11_D', f_bias=1100.0)

Bases: opfunu.cec_based.cec2015.F102015

1

Liang, J. J., Qu, B. Y., & Suganthan, P. N. (2013). Problem definitions and evaluation criteria for the CEC 2014

special session and competition on single objective real-parameter numerical optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou China and Technical Report, Nanyang Technological University, Singapore, 635, 490.

characteristics = []

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 1100.0'

name = 'F11: Hybrid Function 2 (N=4)'

class opfunu.cec_based.cec2015.F12015(ndim=None, bounds=None, f_shift='shift_data_1_D', f_matrix='M_1_D', f_bias=100.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Chen, Q., Liu, B., Zhang, Q., Liang, J., Suganthan, P., & Qu, B. (2014). Problem definitions and evaluation criteria for CEC 2015

special session on bound constrained single-objective computationally expensive numerical optimization. Technical Report, Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou, China and Technical Report, Nanyang Technological University.

characteristics = ['Smooth but narrow ridge']

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 100.0'

linear = False

modality = False

name = 'F1: Rotated Bent Cigar Function'

parametric = True

randomized_term = False

rotated = True

scalable = True

separable = False

shifted = True

unimodal = True

class opfunu.cec_based.cec2015.F122015(ndim=None, bounds=None, f_shift='shift_data_11_D', f_matrix='M_11_D', f_shuffle='shuffle_data_11_D', f_bias=1200.0)

Bases: opfunu.cec_based.cec2015.F102015

1

Liang, J. J., Qu, B. Y., & Suganthan, P. N. (2013). Problem definitions and evaluation criteria for the CEC 2014

special session and competition on single objective real-parameter numerical optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou China and Technical Report, Nanyang Technological University, Singapore, 635, 490.

characteristics = []

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 1200.0'

name = 'F12: Hybrid Function 3 (N=5)'

class opfunu.cec_based.cec2015.F132015(ndim=None, bounds=None, f_shift='shift_data_13_D', f_matrix='M_13_D', f_bias=1300.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Liang, J. J., Qu, B. Y., & Suganthan, P. N. (2013). Problem definitions and evaluation criteria for the CEC 2014

special session and competition on single objective real-parameter numerical optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou China and Technical Report, Nanyang Technological University, Singapore, 635, 490.

characteristics = ['Asymmetrical', 'Different properties around different local optima']

continuous = False

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 1300.0'

linear = False

modality = True

name = 'F13: Composition Function 1 (N=5)'

parametric = True

randomized_term = False

rotated = True

scalable = True

separable = False

shifted = True

unimodal = False

class opfunu.cec_based.cec2015.F142015(ndim=None, bounds=None, f_shift='shift_data_14_D', f_matrix='M_14_D', f_bias=1400.0)

Bases: opfunu.cec_based.cec2015.F132015

1

Liang, J. J., Qu, B. Y., & Suganthan, P. N. (2013). Problem definitions and evaluation criteria for the CEC 2014

special session and competition on single objective real-parameter numerical optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou China and Technical Report, Nanyang Technological University, Singapore, 635, 490.

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 1400.0'

modality = False

name = 'F14: Composition Function 2 (N=3)'

class opfunu.cec_based.cec2015.F152015(ndim=None, bounds=None, f_shift='shift_data_15_D', f_matrix='M_15_D', f_bias=1500.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Liang, J. J., Qu, B. Y., & Suganthan, P. N. (2013). Problem definitions and evaluation criteria for the CEC 2014

special session and competition on single objective real-parameter numerical optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou China and Technical Report, Nanyang Technological University, Singapore, 635, 490.

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 1500.0'

modality = False

name = 'F15: Composition Function 3 (N=5)'

class opfunu.cec_based.cec2015.F22015(ndim=None, bounds=None, f_shift='shift_data_2_D', f_matrix='M_2_D', f_bias=200.0)

Bases: opfunu.cec_based.cec2015.F12015

1

Chen, Q., Liu, B., Zhang, Q., Liang, J., Suganthan, P., & Qu, B. (2014). Problem definitions and evaluation criteria for CEC 2015

special session on bound constrained single-objective computationally expensive numerical optimization. Technical Report, Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou, China and Technical Report, Nanyang Technological University.

characteristics = ['With one sensitive direction']

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 200.0'

name = 'F2: Rotated Discus Function'

class opfunu.cec_based.cec2015.F32015(ndim=None, bounds=None, f_shift='shift_data_3_D', f_matrix='M_3_D', f_bias=300.0)

Bases: opfunu.cec_based.cec2015.F12015

1

Chen, Q., Liu, B., Zhang, Q., Liang, J., Suganthan, P., & Qu, B. (2014). Problem definitions and evaluation criteria for CEC 2015

special session on bound constrained single-objective computationally expensive numerical optimization. Technical Report, Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou, China and Technical Report, Nanyang Technological University.

characteristics = ['Continuous but differentiable only on a set of points']

convex = False

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 300.0'

name = 'F3: Shifted and Rotated Weierstrass Function'

unimodal = False

class opfunu.cec_based.cec2015.F42015(ndim=None, bounds=None, f_shift='shift_data_4_D', f_matrix='M_4_D', f_bias=400.0)

Bases: opfunu.cec_based.cec2015.F12015

1

Chen, Q., Liu, B., Zhang, Q., Liang, J., Suganthan, P., & Qu, B. (2014). Problem definitions and evaluation criteria for CEC 2015

special session on bound constrained single-objective computationally expensive numerical optimization. Technical Report, Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou, China and Technical Report, Nanyang Technological University.

characteristics = ['Local optima’s number is huge', 'The second better local optimum is far from the global optimum']

convex = False

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 400.0'

modality = True

name = 'F4: Shifted and Rotated Schwefel’s Function'

unimodal = False

class opfunu.cec_based.cec2015.F52015(ndim=None, bounds=None, f_shift='shift_data_5_D', f_matrix='M_5_D', f_bias=500.0)

Bases: opfunu.cec_based.cec2015.F12015

1

Chen, Q., Liu, B., Zhang, Q., Liang, J., Suganthan, P., & Qu, B. (2014). Problem definitions and evaluation criteria for CEC 2015

special session on bound constrained single-objective computationally expensive numerical optimization. Technical Report, Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou, China and Technical Report, Nanyang Technological University.

characteristics = ['Continuous everywhere yet differentiable nowhere']

convex = False

differentiable = False

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 500.0'

modality = True

name = 'F5: Shifted and Rotated Katsuura Function'

unimodal = False

class opfunu.cec_based.cec2015.F62015(ndim=None, bounds=None, f_shift='shift_data_6_D', f_matrix='M_6_D', f_bias=600.0)

Bases: opfunu.cec_based.cec2015.F12015

1

Chen, Q., Liu, B., Zhang, Q., Liang, J., Suganthan, P., & Qu, B. (2014). Problem definitions and evaluation criteria for CEC 2015

special session on bound constrained single-objective computationally expensive numerical optimization. Technical Report, Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou, China and Technical Report, Nanyang Technological University.

characteristics = ['Continuous everywhere yet differentiable nowhere']

convex = False

differentiable = False

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 600.0'

name = 'F6: Shifted and Rotated HappyCat Function'

separable = False

unimodal = False

class opfunu.cec_based.cec2015.F72015(ndim=None, bounds=None, f_shift='shift_data_7_D', f_matrix='M_7_D', f_bias=700.0)

Bases: opfunu.cec_based.cec2015.F12015

1

Chen, Q., Liu, B., Zhang, Q., Liang, J., Suganthan, P., & Qu, B. (2014). Problem definitions and evaluation criteria for CEC 2015

special session on bound constrained single-objective computationally expensive numerical optimization. Technical Report, Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou, China and Technical Report, Nanyang Technological University.

characteristics = []

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 700.0'

name = 'F7: Shifted and Rotated HGBat Function'

separable = False

unimodal = False

class opfunu.cec_based.cec2015.F82015(ndim=None, bounds=None, f_shift='shift_data_8_D', f_matrix='M_8_D', f_bias=800.0)

Bases: opfunu.cec_based.cec2015.F12015

1

Chen, Q., Liu, B., Zhang, Q., Liang, J., Suganthan, P., & Qu, B. (2014). Problem definitions and evaluation criteria for CEC 2015

special session on bound constrained single-objective computationally expensive numerical optimization. Technical Report, Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou, China and Technical Report, Nanyang Technological University.

characteristics = []

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 800.0'

name = 'F8: Shifted and Rotated Expanded Griewank’s plus Rosenbrock’s Function'

separable = False

unimodal = False

class opfunu.cec_based.cec2015.F92015(ndim=None, bounds=None, f_shift='shift_data_9_D', f_matrix='M_9_D', f_bias=900.0)

Bases: opfunu.cec_based.cec2015.F12015

1

Chen, Q., Liu, B., Zhang, Q., Liang, J., Suganthan, P., & Qu, B. (2014). Problem definitions and evaluation criteria for CEC 2015

special session on bound constrained single-objective computationally expensive numerical optimization. Technical Report, Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou, China and Technical Report, Nanyang Technological University.

characteristics = []

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 900.0'

modality = True

name = 'F9: Shifted and Rotated Expanded Scaffer’s F6 Function'

separable = False

unimodal = False

opfunu.cec_based.cec2017 module

class opfunu.cec_based.cec2017.F102017(ndim=None, bounds=None, f_shift='shift_data_10', f_matrix='M_10_D', f_shuffle='shuffle_data_10_D', f_bias=1000.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Problem Definitions and Evaluation Criteria for the CEC 2017

Special Session and Competition on Single Objective Real-Parameter Numerical Optimization

characteristics = []

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 1000.0'

linear = False

modality = True

name = 'F10: Hybrid Function 1'

parametric = True

randomized_term = False

rotated = True

scalable = True

separable = False

shifted = True

unimodal = False

class opfunu.cec_based.cec2017.F112017(ndim=None, bounds=None, f_shift='shift_data_11', f_matrix='M_11_D', f_shuffle='shuffle_data_11_D', f_bias=1100.0)

Bases: opfunu.cec_based.cec2017.F102017

1

Problem Definitions and Evaluation Criteria for the CEC 2017

Special Session and Competition on Single Objective Real-Parameter Numerical Optimization

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 1100.0'

name = 'F11: Hybrid Function 2'

class opfunu.cec_based.cec2017.F12017(ndim=None, bounds=None, f_shift='shift_data_1', f_matrix='M_1_D', f_bias=100.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Problem Definitions and Evaluation Criteria for the CEC 2017

Special Session and Competition on Single Objective Real-Parameter Numerical Optimization

characteristics = ['Smooth but narrow ridge']

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 100.0'

linear = False

modality = False

name = 'F1: Shifted and Rotated Bent Cigar'

parametric = True

randomized_term = False

rotated = True

scalable = True

separable = False

shifted = True

unimodal = True

class opfunu.cec_based.cec2017.F122017(ndim=None, bounds=None, f_shift='shift_data_12', f_matrix='M_12_D', f_shuffle='shuffle_data_12_D', f_bias=1200.0)

Bases: opfunu.cec_based.cec2017.F102017

1

Problem Definitions and Evaluation Criteria for the CEC 2017

Special Session and Competition on Single Objective Real-Parameter Numerical Optimization

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 1200.0'

name = 'F12: Hybrid Function 3'

class opfunu.cec_based.cec2017.F132017(ndim=None, bounds=None, f_shift='shift_data_13', f_matrix='M_13_D', f_shuffle='shuffle_data_13_D', f_bias=1300.0)

Bases: opfunu.cec_based.cec2017.F102017

1

Problem Definitions and Evaluation Criteria for the CEC 2017

Special Session and Competition on Single Objective Real-Parameter Numerical Optimization

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 1300.0'

name = 'F13: Hybrid Function 4'

class opfunu.cec_based.cec2017.F142017(ndim=None, bounds=None, f_shift='shift_data_14', f_matrix='M_14_D', f_shuffle='shuffle_data_14_D', f_bias=1400.0)

Bases: opfunu.cec_based.cec2017.F102017

1

Problem Definitions and Evaluation Criteria for the CEC 2017

Special Session and Competition on Single Objective Real-Parameter Numerical Optimization

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 1400.0'

name = 'F14: Hybrid Function 5'

class opfunu.cec_based.cec2017.F152017(ndim=None, bounds=None, f_shift='shift_data_15', f_matrix='M_15_D', f_shuffle='shuffle_data_15_D', f_bias=1500.0)

Bases: opfunu.cec_based.cec2017.F102017

1

Problem Definitions and Evaluation Criteria for the CEC 2017

Special Session and Competition on Single Objective Real-Parameter Numerical Optimization

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 1500.0'

name = 'F15: Hybrid Function 6'

class opfunu.cec_based.cec2017.F162017(ndim=None, bounds=None, f_shift='shift_data_16', f_matrix='M_16_D', f_shuffle='shuffle_data_16_D', f_bias=1600.0)

Bases: opfunu.cec_based.cec2017.F102017

1

Problem Definitions and Evaluation Criteria for the CEC 2017

Special Session and Competition on Single Objective Real-Parameter Numerical Optimization

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 1600.0'

name = 'F16: Hybrid Function 7'

class opfunu.cec_based.cec2017.F172017(ndim=None, bounds=None, f_shift='shift_data_17', f_matrix='M_17_D', f_shuffle='shuffle_data_17_D', f_bias=1700.0)

Bases: opfunu.cec_based.cec2017.F102017

1

Problem Definitions and Evaluation Criteria for the CEC 2017

Special Session and Competition on Single Objective Real-Parameter Numerical Optimization

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 1700.0'

name = 'F17: Hybrid Function 8'

class opfunu.cec_based.cec2017.F182017(ndim=None, bounds=None, f_shift='shift_data_18', f_matrix='M_18_D', f_shuffle='shuffle_data_18_D', f_bias=1800.0)

Bases: opfunu.cec_based.cec2017.F102017

1

Problem Definitions and Evaluation Criteria for the CEC 2017

Special Session and Competition on Single Objective Real-Parameter Numerical Optimization

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 1800.0'

name = 'F18: Hybrid Function 9'

class opfunu.cec_based.cec2017.F192017(ndim=None, bounds=None, f_shift='shift_data_19', f_matrix='M_19_D', f_shuffle='shuffle_data_19_D', f_bias=1900.0)

Bases: opfunu.cec_based.cec2017.F102017

1

Problem Definitions and Evaluation Criteria for the CEC 2017

Special Session and Competition on Single Objective Real-Parameter Numerical Optimization

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 1900.0'

name = 'F19: Hybrid Function 10'

class opfunu.cec_based.cec2017.F202017(ndim=None, bounds=None, f_shift='shift_data_21', f_matrix='M_21_D', f_bias=2000.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Problem Definitions and Evaluation Criteria for the CEC 2017

Special Session and Competition on Single Objective Real-Parameter Numerical Optimization

characteristics = ['Asymmetrical', 'Different properties around different local optima']

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 2000.0'

linear = False

modality = False

name = 'F20: Composition Function 1'

parametric = True

randomized_term = False

rotated = True

scalable = True

separable = False

shifted = True

unimodal = False

class opfunu.cec_based.cec2017.F212017(ndim=None, bounds=None, f_shift='shift_data_22', f_matrix='M_22_D', f_bias=2100.0)

Bases: opfunu.cec_based.cec2017.F202017

1

Problem Definitions and Evaluation Criteria for the CEC 2017

Special Session and Competition on Single Objective Real-Parameter Numerical Optimization

characteristics = ['Asymmetrical', 'Different properties around different local optima']

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 2100.0'

modality = True

name = 'F21: Composition Function 2'

class opfunu.cec_based.cec2017.F22017(ndim=None, bounds=None, f_shift='shift_data_2', f_matrix='M_2_D', f_bias=200.0)

Bases: opfunu.cec_based.cec2017.F12017

1

Problem Definitions and Evaluation Criteria for the CEC 2017

Special Session and Competition on Single Objective Real-Parameter Numerical Optimization

characteristics = []

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 200.0'

name = 'F2: Shifted and Rotated Zakharov Function'

class opfunu.cec_based.cec2017.F222017(ndim=None, bounds=None, f_shift='shift_data_23', f_matrix='M_23_D', f_bias=2200.0)

Bases: opfunu.cec_based.cec2017.F202017

1

Problem Definitions and Evaluation Criteria for the CEC 2017

Special Session and Competition on Single Objective Real-Parameter Numerical Optimization

characteristics = ['Asymmetrical', 'Different properties around different local optima']

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 2200.0'

modality = True

name = 'F22: Composition Function 3'

class opfunu.cec_based.cec2017.F232017(ndim=None, bounds=None, f_shift='shift_data_24', f_matrix='M_24_D', f_bias=2300.0)

Bases: opfunu.cec_based.cec2017.F202017

1

Problem Definitions and Evaluation Criteria for the CEC 2017

Special Session and Competition on Single Objective Real-Parameter Numerical Optimization

characteristics = ['Asymmetrical', 'Different properties around different local optima']

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 2300.0'

modality = True

name = 'F23: Composition Function 4'

class opfunu.cec_based.cec2017.F242017(ndim=None, bounds=None, f_shift='shift_data_25', f_matrix='M_25_D', f_bias=2400.0)

Bases: opfunu.cec_based.cec2017.F202017

1

Problem Definitions and Evaluation Criteria for the CEC 2017

Special Session and Competition on Single Objective Real-Parameter Numerical Optimization

characteristics = ['Asymmetrical', 'Different properties around different local optima']

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 2400.0'

modality = True

name = 'F24: Composition Function 5'

class opfunu.cec_based.cec2017.F252017(ndim=None, bounds=None, f_shift='shift_data_26', f_matrix='M_26_D', f_bias=2500.0)

Bases: opfunu.cec_based.cec2017.F202017

1

Problem Definitions and Evaluation Criteria for the CEC 2017

Special Session and Competition on Single Objective Real-Parameter Numerical Optimization

characteristics = ['Asymmetrical', 'Different properties around different local optima']

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 2500.0'

modality = True

name = 'F25: Composition Function 6'

class opfunu.cec_based.cec2017.F262017(ndim=None, bounds=None, f_shift='shift_data_27', f_matrix='M_27_D', f_bias=2600.0)

Bases: opfunu.cec_based.cec2017.F202017

1

Problem Definitions and Evaluation Criteria for the CEC 2017

Special Session and Competition on Single Objective Real-Parameter Numerical Optimization

characteristics = ['Asymmetrical', 'Different properties around different local optima']

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 2600.0'

modality = True

name = 'F26: Composition Function 7'

class opfunu.cec_based.cec2017.F272017(ndim=None, bounds=None, f_shift='shift_data_28', f_matrix='M_28_D', f_bias=2700.0)

Bases: opfunu.cec_based.cec2017.F202017

1

Problem Definitions and Evaluation Criteria for the CEC 2017

Special Session and Competition on Single Objective Real-Parameter Numerical Optimization

characteristics = ['Asymmetrical', 'Different properties around different local optima']

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 2700.0'

name = 'F27: Composition Function 8'

class opfunu.cec_based.cec2017.F282017(ndim=None, bounds=None, f_shift='shift_data_29', f_matrix='M_29_D', f_shuffle='shuffle_data_29_D', f_bias=2800.0)

Bases: opfunu.cec_based.cec2017.F202017

1

Problem Definitions and Evaluation Criteria for the CEC 2017

Special Session and Competition on Single Objective Real-Parameter Numerical Optimization

characteristics = ['Asymmetrical', 'Different properties around different local optima', 'Different properties for different variables subcomponents']

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 2800.0'

name = 'F28: Composition Function 9'

class opfunu.cec_based.cec2017.F292017(ndim=None, bounds=None, f_shift='shift_data_30', f_matrix='M_30_D', f_shuffle='shuffle_data_30_D', f_bias=2900.0)

Bases: opfunu.cec_based.cec2017.F202017

1

Problem Definitions and Evaluation Criteria for the CEC 2017

Special Session and Competition on Single Objective Real-Parameter Numerical Optimization

characteristics = ['Asymmetrical', 'Different properties around different local optima', 'Different properties for different variables subcomponents']

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 2900.0'

name = 'F29: Composition Function 10'

class opfunu.cec_based.cec2017.F32017(ndim=None, bounds=None, f_shift='shift_data_3', f_matrix='M_3_D', f_bias=300.0)

Bases: opfunu.cec_based.cec2017.F12017

1

Problem Definitions and Evaluation Criteria for the CEC 2017

Special Session and Competition on Single Objective Real-Parameter Numerical Optimization

characteristics = ['Local optima’s number is huge']

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 300.0'

modality = True

name = 'F3: Shifted and Rotated Rosenbrock’s Function'

unimodal = False

class opfunu.cec_based.cec2017.F42017(ndim=None, bounds=None, f_shift='shift_data_4', f_matrix='M_4_D', f_bias=400.0)

Bases: opfunu.cec_based.cec2017.F12017

1

Problem Definitions and Evaluation Criteria for the CEC 2017

Special Session and Competition on Single Objective Real-Parameter Numerical Optimization

characteristics = ['Local optima’s number is huge', 'The second better local optimum is far from the global optimum']

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 400.0'

modality = True

name = 'F4: Shifted and Rotated Rastrigin’s Function'

unimodal = False

class opfunu.cec_based.cec2017.F52017(ndim=None, bounds=None, f_shift='shift_data_5', f_matrix='M_5_D', f_bias=500.0)

Bases: opfunu.cec_based.cec2017.F12017

1

Problem Definitions and Evaluation Criteria for the CEC 2017

Special Session and Competition on Single Objective Real-Parameter Numerical Optimization

characteristics = ['Asymmetrical', 'Local optima’s number is huge']

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 500.0'

modality = True

name = 'F5: Shifted and Rotated Schaffer’s F7 Function'

unimodal = False

class opfunu.cec_based.cec2017.F62017(ndim=None, bounds=None, f_shift='shift_data_6', f_matrix='M_6_D', f_bias=600.0)

Bases: opfunu.cec_based.cec2017.F12017

1

Problem Definitions and Evaluation Criteria for the CEC 2017

Special Session and Competition on Single Objective Real-Parameter Numerical Optimization

characteristics = ['Asymmetrical', 'Continuous everywhere yet differentiable nowhere']

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 600.0'

modality = True

name = 'F6: Shifted and Rotated Lunacek Bi-Rastrigin’s Function'

unimodal = False

class opfunu.cec_based.cec2017.F72017(ndim=None, bounds=None, f_shift='shift_data_7', f_matrix='M_7_D', f_bias=700.0)

Bases: opfunu.cec_based.cec2017.F12017

1

Problem Definitions and Evaluation Criteria for the CEC 2017

Special Session and Competition on Single Objective Real-Parameter Numerical Optimization

characteristics = ['Asymmetrical', 'Local optima’s number is huge']

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 700.0'

modality = True

name = 'F7: Shifted and Rotated Non-Continuous Rastrigin’s Function'

unimodal = False

class opfunu.cec_based.cec2017.F82017(ndim=None, bounds=None, f_shift='shift_data_8', f_matrix='M_8_D', f_bias=800.0)

Bases: opfunu.cec_based.cec2017.F12017

1

Problem Definitions and Evaluation Criteria for the CEC 2017

Special Session and Competition on Single Objective Real-Parameter Numerical Optimization

characteristics = ['Local optima’s number is huge']

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 800.0'

modality = True

name = 'F8: Shifted and Rotated Levy Function'

unimodal = False

class opfunu.cec_based.cec2017.F92017(ndim=None, bounds=None, f_shift='shift_data_9', f_matrix='M_9_D', f_bias=900.0)

Bases: opfunu.cec_based.cec2017.F12017

1

Problem Definitions and Evaluation Criteria for the CEC 2017

Special Session and Competition on Single Objective Real-Parameter Numerical Optimization

characteristics = ['Local optima’s number is huge', 'The second better local optimum is far from the global optimum']

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 900.0'

modality = True

name = 'F9: Shifted and Rotated Schwefel’s Function'

unimodal = False

opfunu.cec_based.cec2019 module

class opfunu.cec_based.cec2019.F102019(ndim=None, bounds=None, f_shift='shift_data_10', f_matrix='M_10_D', f_bias=1.0)

Bases: opfunu.cec_based.cec2019.F42019

1

The 100-Digit Challenge: Problem Definitions and Evaluation Criteria for the 100-Digit

Challenge Special Session and Competition on Single Objective Numerical Optimization

characteristics = []

convex = False

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 1.0'

modality = False

name = 'F10: Shifted and Rotated Ackley Function'

class opfunu.cec_based.cec2019.F12019(ndim=None, bounds=None, f_shift='shift_data_1', f_bias=1.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

The 100-Digit Challenge: Problem Definitions and Evaluation Criteria for the 100-Digit

Challenge Special Session and Competition on Single Objective Numerical Optimization

characteristics = ['Multimodal with one global minimum', 'Very highly conditioned', 'fully parameter-dependent']

continuous = False

convex = False

differentiable = False

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 1.0'

linear = False

modality = True

name = 'F1: Storn’s Chebyshev Polynomial Fitting Problem'

parametric = True

randomized_term = False

rotated = True

scalable = False

separable = False

shifted = True

unimodal = False

class opfunu.cec_based.cec2019.F22019(ndim=None, bounds=None, f_shift='shift_data_2', f_bias=1.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

The 100-Digit Challenge: Problem Definitions and Evaluation Criteria for the 100-Digit

Challenge Special Session and Competition on Single Objective Numerical Optimization

characteristics = ['Multimodal with one global minimum', 'Very highly conditioned', 'fully parameter-dependent']

continuous = False

convex = False

differentiable = False

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 1.0'

linear = False

modality = True

name = 'F2: Inverse Hilbert Matrix Problem'

parametric = True

randomized_term = False

rotated = True

scalable = False

separable = False

shifted = True

unimodal = False

class opfunu.cec_based.cec2019.F32019(ndim=None, bounds=None, f_shift='shift_data_3', f_bias=1.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

The 100-Digit Challenge: Problem Definitions and Evaluation Criteria for the 100-Digit

Challenge Special Session and Competition on Single Objective Numerical Optimization

**Note: The CEC 2019 implementation and this implementation results match when x* = [0,…,0] and

characteristics = ['Multimodal with one global minimum', 'Very highly conditioned', 'fully parameter-dependent']

continuous = False

convex = False

differentiable = False

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 1.0'

linear = False

modality = True

name = 'F3: Lennard-Jones Minimum Energy Cluster Problem'

parametric = True

randomized_term = False

rotated = True

scalable = False

separable = False

shifted = True

unimodal = False

class opfunu.cec_based.cec2019.F42019(ndim=None, bounds=None, f_shift='shift_data_4', f_matrix='M_1_D', f_bias=1.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

The 100-Digit Challenge: Problem Definitions and Evaluation Criteria for the 100-Digit

Challenge Special Session and Competition on Single Objective Numerical Optimization

characteristics = ['Local optima’s number is huge', 'The penultimate optimum is far from the global optimum']

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 1.0'

linear = False

modality = True

name = 'F4: Shifted and Rotated Rastrigin’s Function'

parametric = True

randomized_term = False

rotated = True

scalable = True

separable = False

shifted = True

unimodal = False

class opfunu.cec_based.cec2019.F52019(ndim=None, bounds=None, f_shift='shift_data_5', f_matrix='M_5_D', f_bias=1.0)

Bases: opfunu.cec_based.cec2019.F42019

1

The 100-Digit Challenge: Problem Definitions and Evaluation Criteria for the 100-Digit

Challenge Special Session and Competition on Single Objective Numerical Optimization

characteristics = []

convex = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 1.0'

modality = False

name = 'F5: Shifted and Rotated Griewank’s Function'

class opfunu.cec_based.cec2019.F62019(ndim=None, bounds=None, f_shift='shift_data_6', f_matrix='M_6_D', f_bias=1.0)

Bases: opfunu.cec_based.cec2019.F42019

1

The 100-Digit Challenge: Problem Definitions and Evaluation Criteria for the 100-Digit

Challenge Special Session and Competition on Single Objective Numerical Optimization

characteristics = ['Local optima’s number is huge']

convex = False

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 1.0'

modality = True

name = 'F6: Shifted and Rotated Weierstrass Function'

class opfunu.cec_based.cec2019.F72019(ndim=None, bounds=None, f_shift='shift_data_7', f_matrix='M_7_D', f_bias=1.0)

Bases: opfunu.cec_based.cec2019.F42019

1

The 100-Digit Challenge: Problem Definitions and Evaluation Criteria for the 100-Digit

Challenge Special Session and Competition on Single Objective Numerical Optimization

characteristics = ['Local optima’s number is huge']

convex = False

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 1.0'

modality = True

name = 'F7: Shifted and Rotated Schwefel’s Function'

class opfunu.cec_based.cec2019.F82019(ndim=None, bounds=None, f_shift='shift_data_8', f_matrix='M_8_D', f_bias=1.0)

Bases: opfunu.cec_based.cec2019.F42019

1

The 100-Digit Challenge: Problem Definitions and Evaluation Criteria for the 100-Digit

Challenge Special Session and Competition on Single Objective Numerical Optimization

characteristics = ['Local optima’s number is huge']

convex = False

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 1.0'

modality = True

name = 'F8: Shifted and Rotated Expanded Schaffer’s F6 Function'

class opfunu.cec_based.cec2019.F92019(ndim=None, bounds=None, f_shift='shift_data_9', f_matrix='M_9_D', f_bias=1.0)

Bases: opfunu.cec_based.cec2019.F42019

1

The 100-Digit Challenge: Problem Definitions and Evaluation Criteria for the 100-Digit

Challenge Special Session and Competition on Single Objective Numerical Optimization

characteristics = []

convex = False

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 1.0'

modality = False

name = 'F9: Shifted and Rotated Happy Cat Function'

opfunu.cec_based.cec2020 module

class opfunu.cec_based.cec2020.F102020(ndim=None, bounds=None, f_shift='shift_data_25', f_matrix='M_25_D', f_bias=2500.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Problem Definitions and Evaluation Criteria for the CEC 2020

Special Session and Competition on Single Objective Bound Constrained Numerical Optimization

characteristics = ['Asymmetrical', 'Different properties around different local optima']

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 2500.0'

linear = False

modality = True

name = 'F10: Composition Function 3 (F24 CEC-2017)'

parametric = True

randomized_term = False

rotated = True

scalable = True

separable = False

shifted = True

unimodal = False

class opfunu.cec_based.cec2020.F12020(ndim=None, bounds=None, f_shift='shift_data_1', f_matrix='M_1_D', f_bias=100.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Problem Definitions and Evaluation Criteria for the CEC 2020

Special Session and Competition on Single Objective Bound Constrained Numerical Optimization

characteristics = ['Smooth but narrow ridge']

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 100.0'

linear = False

modality = False

name = 'F1: Shifted and Rotated Bent Cigar Function (F1 CEC-2017)'

parametric = True

randomized_term = False

rotated = True

scalable = True

separable = False

shifted = True

unimodal = True

class opfunu.cec_based.cec2020.F22020(ndim=None, bounds=None, f_shift='shift_data_2', f_matrix='M_2_D', f_bias=1100.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Problem Definitions and Evaluation Criteria for the CEC 2020

Special Session and Competition on Single Objective Bound Constrained Numerical Optimization

characteristics = ['Local optima’s number is huge', 'The penultimate local optimum is far from the global optimum.']

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 1100.0'

linear = False

modality = True

name = 'F2: Shifted and Rotated Schwefel’s Function (F11 CEC-2014)'

parametric = True

randomized_term = False

rotated = True

scalable = True

separable = False

shifted = True

unimodal = False

class opfunu.cec_based.cec2020.F32020(ndim=None, bounds=None, f_shift='shift_data_3', f_matrix='M_3_D', f_bias=700.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Problem Definitions and Evaluation Criteria for the CEC 2020

Special Session and Competition on Single Objective Bound Constrained Numerical Optimization

characteristics = ['Asymmetrical', 'Continuous everywhere yet differentiable nowhere']

continuous = True

convex = False

differentiable = False

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 700.0'

linear = False

modality = False

name = 'F3: Shifted and Rotated Lunacek bi-Rastrigin Function (F7 CEC-2017)'

parametric = True

randomized_term = False

rotated = True

scalable = True

separable = False

shifted = True

unimodal = False

class opfunu.cec_based.cec2020.F42020(ndim=None, bounds=None, f_shift='shift_data_4', f_matrix='M_4_D', f_bias=1900.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Problem Definitions and Evaluation Criteria for the CEC 2020

Special Session and Competition on Single Objective Bound Constrained Numerical Optimization

characteristics = ['Optimal point locates in flat area']

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 1900.0'

linear = False

modality = False

name = 'F4: Expanded Rosenbrock’s plus Griewank’s Function (F15 CEC-2014)'

parametric = True

randomized_term = False

rotated = True

scalable = True

separable = False

shifted = True

unimodal = False

class opfunu.cec_based.cec2020.F52020(ndim=None, bounds=None, f_shift='shift_data_6', f_matrix='M_6_D', f_shuffle='shuffle_data_6_D', f_bias=1700.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Problem Definitions and Evaluation Criteria for the CEC 2020

Special Session and Competition on Single Objective Bound Constrained Numerical Optimization

characteristics = ['Different properties for different variables subcomponents']

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 1700.0'

linear = False

modality = True

name = 'F17: Hybrid Function 1 (F17 CEC-2014)'

parametric = True

randomized_term = False

rotated = True

scalable = True

separable = False

shifted = True

unimodal = False

class opfunu.cec_based.cec2020.F62020(ndim=None, bounds=None, f_shift='shift_data_7', f_matrix='M_7_D', f_shuffle='shuffle_data_7_D', f_bias=1600.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Problem Definitions and Evaluation Criteria for the CEC 2020

Special Session and Competition on Single Objective Bound Constrained Numerical Optimization

characteristics = []

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 1600.0'

linear = False

modality = True

name = 'F16: Hybrid Function 2 (F15 CEC-2017)'

parametric = True

randomized_term = False

rotated = True

scalable = True

separable = False

shifted = True

unimodal = False

class opfunu.cec_based.cec2020.F72020(ndim=None, bounds=None, f_shift='shift_data_16', f_matrix='M_16_D', f_shuffle='shuffle_data_16_D', f_bias=2100.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Problem Definitions and Evaluation Criteria for the CEC 2020

Special Session and Competition on Single Objective Bound Constrained Numerical Optimization

characteristics = ['Different properties for different variables subcomponents']

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 2100.0'

linear = False

modality = True

name = 'F7: Hybrid Function 3 (F21 CEC-2014)'

parametric = True

randomized_term = False

rotated = True

scalable = True

separable = False

shifted = True

unimodal = False

class opfunu.cec_based.cec2020.F82020(ndim=None, bounds=None, f_shift='shift_data_22', f_matrix='M_22_D', f_bias=2200.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Problem Definitions and Evaluation Criteria for the CEC 2020

Special Session and Competition on Single Objective Bound Constrained Numerical Optimization

characteristics = ['Asymmetrical', 'Different properties around different local optima']

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 2200.0'

linear = False

modality = True

name = 'F8: Composition Function 1 (F21 CEC-2017)'

parametric = True

randomized_term = False

rotated = True

scalable = True

separable = False

shifted = True

unimodal = False

class opfunu.cec_based.cec2020.F92020(ndim=None, bounds=None, f_shift='shift_data_24', f_matrix='M_24_D', f_bias=2400.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Problem Definitions and Evaluation Criteria for the CEC 2020

Special Session and Competition on Single Objective Bound Constrained Numerical Optimization

characteristics = ['Asymmetrical', 'Different properties around different local optima']

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 2400.0'

linear = False

modality = True

name = 'F9: Composition Function 2 (F23 CEC-2017)'

parametric = True

randomized_term = False

rotated = True

scalable = True

separable = False

shifted = True

unimodal = False

opfunu.cec_based.cec2021 module

class opfunu.cec_based.cec2021.F102021(ndim=None, bounds=None, f_shift='shift_data_10', f_matrix='M_10_D', f_bias=2500.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Problem Definitions and Evaluation Criteria for the CEC 2021

Special Session and Competition on Single Objective Bound Constrained Numerical Optimization

characteristics = ['Asymmetrical', 'Different properties around different local optima']

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 2500.0'

linear = False

modality = True

name = 'F10: Composition Function 3 (F24 CEC-2017)'

parametric = True

randomized_term = False

rotated = True

scalable = True

separable = False

shifted = True

unimodal = False

class opfunu.cec_based.cec2021.F12021(ndim=None, bounds=None, f_shift='shift_data_1', f_matrix='M_1_D', f_bias=100.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Problem Definitions and Evaluation Criteria for the CEC 2021

Special Session and Competition on Single Objective Bound Constrained Numerical Optimization

characteristics = ['Smooth but narrow ridge']

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 100.0'

linear = False

modality = False

name = 'F1: Shifted and Rotated Bent Cigar Function (F1 CEC-2017)'

parametric = True

randomized_term = False

rotated = True

scalable = True

separable = False

shifted = True

unimodal = True

class opfunu.cec_based.cec2021.F22021(ndim=None, bounds=None, f_shift='shift_data_2', f_matrix='M_2_D', f_bias=1100.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Problem Definitions and Evaluation Criteria for the CEC 2021

Special Session and Competition on Single Objective Bound Constrained Numerical Optimization

characteristics = ['Local optima’s number is huge', 'The penultimate local optimum is far from the global optimum.']

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 1100.0'

linear = False

modality = True

name = 'F2: Shifted and Rotated Schwefel’s Function (F11 CEC-2014)'

parametric = True

randomized_term = False

rotated = True

scalable = True

separable = False

shifted = True

unimodal = False

class opfunu.cec_based.cec2021.F32021(ndim=None, bounds=None, f_shift='shift_data_3', f_matrix='M_3_D', f_bias=700.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Problem Definitions and Evaluation Criteria for the CEC 2021

Special Session and Competition on Single Objective Bound Constrained Numerical Optimization

characteristics = ['Asymmetrical', 'Continuous everywhere yet differentiable nowhere']

continuous = True

convex = False

differentiable = False

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 700.0'

linear = False

modality = False

name = 'F3: Shifted and Rotated Lunacek bi-Rastrigin Function (F7 CEC-2017)'

parametric = True

randomized_term = False

rotated = True

scalable = True

separable = False

shifted = True

unimodal = False

class opfunu.cec_based.cec2021.F42021(ndim=None, bounds=None, f_shift='shift_data_4', f_matrix='M_4_D', f_bias=1900.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Problem Definitions and Evaluation Criteria for the CEC 2021

Special Session and Competition on Single Objective Bound Constrained Numerical Optimization

characteristics = ['Optimal point locates in flat area']

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 1900.0'

linear = False

modality = False

name = 'F4: Expanded Rosenbrock’s plus Griewangk’s Function (F15 CEC-2014)'

parametric = True

randomized_term = False

rotated = True

scalable = True

separable = False

shifted = True

unimodal = False

class opfunu.cec_based.cec2021.F52021(ndim=None, bounds=None, f_shift='shift_data_5', f_matrix='M_5_D', f_shuffle='shuffle_data_5_D', f_bias=1700.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Problem Definitions and Evaluation Criteria for the CEC 2021

Special Session and Competition on Single Objective Bound Constrained Numerical Optimization

characteristics = ['Different properties for different variables subcomponents']

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 1700.0'

linear = False

modality = True

name = 'F17: Hybrid Function 1 (F17 CEC-2014)'

parametric = True

randomized_term = False

rotated = True

scalable = True

separable = False

shifted = True

unimodal = False

class opfunu.cec_based.cec2021.F62021(ndim=None, bounds=None, f_shift='shift_data_6', f_matrix='M_6_D', f_shuffle='shuffle_data_6_D', f_bias=1600.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Problem Definitions and Evaluation Criteria for the CEC 2021

Special Session and Competition on Single Objective Bound Constrained Numerical Optimization

characteristics = []

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 1600.0'

linear = False

modality = True

name = 'F16: Hybrid Function 2 (F15 CEC-2017)'

parametric = True

randomized_term = False

rotated = True

scalable = True

separable = False

shifted = True

unimodal = False

class opfunu.cec_based.cec2021.F72021(ndim=None, bounds=None, f_shift='shift_data_7', f_matrix='M_7_D', f_shuffle='shuffle_data_7_D', f_bias=2100.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Problem Definitions and Evaluation Criteria for the CEC 2021

Special Session and Competition on Single Objective Bound Constrained Numerical Optimization

characteristics = ['Different properties for different variables subcomponents']

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 2100.0'

linear = False

modality = True

name = 'F7: Hybrid Function 3 (F21 CEC-2014)'

parametric = True

randomized_term = False

rotated = True

scalable = True

separable = False

shifted = True

unimodal = False

class opfunu.cec_based.cec2021.F82021(ndim=None, bounds=None, f_shift='shift_data_8', f_matrix='M_8_D', f_bias=2200.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Problem Definitions and Evaluation Criteria for the CEC 2021

Special Session and Competition on Single Objective Bound Constrained Numerical Optimization

characteristics = ['Asymmetrical', 'Different properties around different local optima']

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 2200.0'

linear = False

modality = True

name = 'F8: Composition Function 1 (F21 CEC-2017)'

parametric = True

randomized_term = False

rotated = True

scalable = True

separable = False

shifted = True

unimodal = False

class opfunu.cec_based.cec2021.F92021(ndim=None, bounds=None, f_shift='shift_data_9', f_matrix='M_9_D', f_bias=2400.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Problem Definitions and Evaluation Criteria for the CEC 2021

Special Session and Competition on Single Objective Bound Constrained Numerical Optimization

characteristics = ['Asymmetrical', 'Different properties around different local optima']

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 2400.0'

linear = False

modality = True

name = 'F9: Composition Function 2 (F23 CEC-2017)'

parametric = True

randomized_term = False

rotated = True

scalable = True

separable = False

shifted = True

unimodal = False

opfunu.cec_based.cec2022 module

class opfunu.cec_based.cec2022.F102022(ndim=None, bounds=None, f_shift='shift_data_10', f_matrix='M_10_D', f_bias=2400.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Problem Definitions and Evaluation Criteria for the CEC 2022

Special Session and Competition on Single Objective Bound Constrained Numerical Optimization

characteristics = ['Asymmetrical', 'Different properties around different local optima']

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 2400.0'

linear = False

modality = True

name = 'F10: Composition Function 2'

parametric = True

randomized_term = False

rotated = True

scalable = True

separable = False

shifted = True

unimodal = False

class opfunu.cec_based.cec2022.F112022(ndim=None, bounds=None, f_shift='shift_data_11', f_matrix='M_11_D', f_bias=2600.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Problem Definitions and Evaluation Criteria for the CEC 2022

Special Session and Competition on Single Objective Bound Constrained Numerical Optimization

characteristics = ['Asymmetrical', 'Different properties around different local optima']

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 2600.0'

linear = False

modality = True

name = 'F11: Composition Function 3'

parametric = True

randomized_term = False

rotated = True

scalable = True

separable = False

shifted = True

unimodal = False

class opfunu.cec_based.cec2022.F12022(ndim=None, bounds=None, f_shift='shift_data_1', f_matrix='M_1_D', f_bias=300.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Problem Definitions and Evaluation Criteria for the CEC 2022

Special Session and Competition on Single Objective Bound Constrained Numerical Optimization

characteristics = []

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 300.0'

linear = False

modality = False

name = 'F1: Shifted and full Rotated Zakharov Function'

parametric = True

randomized_term = False

rotated = True

scalable = True

separable = False

shifted = True

unimodal = True

class opfunu.cec_based.cec2022.F122022(ndim=None, bounds=None, f_shift='shift_data_12', f_matrix='M_12_D', f_bias=2700.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Problem Definitions and Evaluation Criteria for the CEC 2022

Special Session and Competition on Single Objective Bound Constrained Numerical Optimization

characteristics = ['Asymmetrical', 'Different properties around different local optima']

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 2700.0'

linear = False

modality = True

name = 'F12: Composition Function 4'

parametric = True

randomized_term = False

rotated = True

scalable = True

separable = False

shifted = True

unimodal = False

class opfunu.cec_based.cec2022.F22022(ndim=None, bounds=None, f_shift='shift_data_2', f_matrix='M_2_D', f_bias=400.0)

Bases: opfunu.cec_based.cec2022.F12022

1

Problem Definitions and Evaluation Criteria for the CEC 2022

Special Session and Competition on Single Objective Bound Constrained Numerical Optimization

characteristics = ['Local optima’s number is huge']

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 400.0'

modality = True

name = 'F2: Shifted and Rotated Rosenbrock’s Function'

unimodal = False

class opfunu.cec_based.cec2022.F32022(ndim=None, bounds=None, f_shift='shift_data_3', f_matrix='M_3_D', f_bias=600.0)

Bases: opfunu.cec_based.cec2022.F12022

1

Problem Definitions and Evaluation Criteria for the CEC 2022

Special Session and Competition on Single Objective Bound Constrained Numerical Optimization

characteristics = ['Asymmetrical', 'Local optima’s number is huge']

convex = False

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 600.0'

modality = True

name = 'F3: Shifted and full Rotated Expanded Schaffer’s F7'

unimodal = False

class opfunu.cec_based.cec2022.F42022(ndim=None, bounds=None, f_shift='shift_data_4', f_matrix='M_4_D', f_bias=800.0)

Bases: opfunu.cec_based.cec2022.F12022

1

Problem Definitions and Evaluation Criteria for the CEC 2022

Special Session and Competition on Single Objective Bound Constrained Numerical Optimization

characteristics = ['Asymmetrical', 'Local optima’s number is huge']

convex = False

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 800.0'

modality = True

name = 'F4: Shifted and Rotated Non-Continuous Rastrigin’s Function'

unimodal = False

class opfunu.cec_based.cec2022.F52022(ndim=None, bounds=None, f_shift='shift_data_5', f_matrix='M_5_D', f_bias=900.0)

Bases: opfunu.cec_based.cec2022.F12022

1

Problem Definitions and Evaluation Criteria for the CEC 2022

Special Session and Competition on Single Objective Bound Constrained Numerical Optimization

characteristics = ['Local optima’s number is huge']

convex = False

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 900.0'

modality = True

name = 'F5: Shifted and Rotated Levy Function'

unimodal = False

class opfunu.cec_based.cec2022.F62022(ndim=None, bounds=None, f_shift='shift_data_6', f_matrix='M_6_D', f_shuffle='shuffle_data_6_D', f_bias=1800.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Problem Definitions and Evaluation Criteria for the CEC 2022

Special Session and Competition on Single Objective Bound Constrained Numerical Optimization

characteristics = ['Different properties for different variables subcomponents']

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 1800.0'

linear = False

modality = True

name = 'F6: Hybrid Function 1'

parametric = True

randomized_term = False

rotated = True

scalable = True

separable = False

shifted = True

unimodal = False

class opfunu.cec_based.cec2022.F72022(ndim=None, bounds=None, f_shift='shift_data_7', f_matrix='M_7_D', f_shuffle='shuffle_data_7_D', f_bias=2000.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Problem Definitions and Evaluation Criteria for the CEC 2022

Special Session and Competition on Single Objective Bound Constrained Numerical Optimization

characteristics = ['Different properties for different variables subcomponents']

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 2000.0'

linear = False

modality = True

name = 'F7: Hybrid Function 2'

parametric = True

randomized_term = False

rotated = True

scalable = True

separable = False

shifted = True

unimodal = False

class opfunu.cec_based.cec2022.F82022(ndim=None, bounds=None, f_shift='shift_data_8', f_matrix='M_8_D', f_shuffle='shuffle_data_8_D', f_bias=2200.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Problem Definitions and Evaluation Criteria for the CEC 2022

Special Session and Competition on Single Objective Bound Constrained Numerical Optimization

characteristics = ['Different properties for different variables subcomponents']

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 2200.0'

linear = False

modality = True

name = 'F8: Hybrid Function 3'

parametric = True

randomized_term = False

rotated = True

scalable = True

separable = False

shifted = True

unimodal = False

class opfunu.cec_based.cec2022.F92022(ndim=None, bounds=None, f_shift='shift_data_9', f_matrix='M_9_D', f_bias=2300.0)

Bases: opfunu.cec_based.cec.CecBenchmark

1

Problem Definitions and Evaluation Criteria for the CEC 2022

Special Session and Competition on Single Objective Bound Constrained Numerical Optimization

characteristics = ['Asymmetrical', 'Different properties around different local optima']

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'F_1(x) = \\sum_{i=1}^D z_i^2 + bias, z=x-o,\\\\ x=[x_1, ..., x_D]; o=[o_1, ..., o_D]: \\text{the shifted global optimum}'

latex_formula_bounds = 'x_i \\in [-100.0, 100.0], \\forall i \\in  [1, D]'

latex_formula_dimension = '2 <= D <= 100'

latex_formula_global_optimum = '\\text{Global optimum: } x^* = o, F_1(x^*) = bias = 2300.0'

linear = False

modality = True

name = 'F9: Composition Function 1'

parametric = True

randomized_term = False

rotated = True

scalable = True

separable = False

shifted = True

unimodal = False

opfunu.name_based package

opfunu.name_based.a_func module

class opfunu.name_based.a_func.AMGM(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

[1] The AMGM (Arithmetic Mean - Geometric Mean Equality). [2] Gavana, A. Global Optimization Benchmarks and AMPGO, retrieved 2015

continuous = True

convex = True

differentiable = False

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f_{\\text{AMGM}}(x) = \\left ( \\frac{1}{n} \\sum_{i=1}^{n} x_i - \\sqrt[n]{ \\prod_{i=1}^{n} x_i} \\right )^2'

latex_formula_bounds = 'x_i \\in [0., 10.0], \\forall i \\in \\llbracket 1, d\\rrbracket'

latex_formula_dimension = 'd \\in \\mathbb{N}_{+}^{*}'

latex_formula_global_optimum = 'f(x*)\\approx 0, at$$ $$x_1=x_2=...=x_n'

linear = False

modality = False

name = 'AMGM Function'

parametric = False

randomized_term = False

scalable = True

separable = False

unimodal = False

class opfunu.name_based.a_func.Ackley01(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Adorio, E. MVF - “Multivariate Test Functions Library in C for Unconstrained Global Optimization”, 2005

TODO: the -0.2 factor in the exponent of the first term is given as -0.02 in Jamil et al.

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f_{\\text{Ackley01}}(x) = -20 e^{-0.2 \\sqrt{\\frac{1}{n} \\sum_{i=1}^n x_i^2}} - e^{\\frac{1}{n} \\sum_{i=1}^n \\cos(2 \\pi x_i)} + 20 + e'

latex_formula_bounds = 'x_i \\in [-35, 35], \\forall i \\in \\llbracket 1, d\\rrbracket'

latex_formula_dimension = 'd \\in \\mathbb{N}_{+}^{*}'

latex_formula_global_optimum = 'f(0, ..., 0) = 0'

linear = False

modality = False

name = 'Ackley 01'

parametric = False

randomized_term = False

scalable = True

separable = False

unimodal = False

class opfunu.name_based.a_func.Ackley02(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark FunctionsFor Global Optimization Problems Int.

Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f_{\\text{Ackley02}(x) = -200 e^{-0.02 \\sqrt{x_1^2 + x_2^2}}'

latex_formula_bounds = 'x_i \\in [-32.0, 32.0], \\forall i \\in \\llbracket 1, d\\rrbracket'

latex_formula_dimension = 'd = 2'

latex_formula_global_optimum = 'f(0, ..., 0) = -200'

linear = False

modality = False

name = 'Ackley 02'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = True

class opfunu.name_based.a_func.Ackley03(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark FunctionsFor Global Optimization Problems Int.

Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f_{\\text{Ackley03}}(x) = -200 e^{-0.02 \\sqrt{x_1^2 + x_2^2}} + 5e^{\\cos(3x_1) + \\sin(3x_2)}'

latex_formula_bounds = 'x_i \\in [-32.0, 32.0], \\forall i \\in \\llbracket 1, d\\rrbracket'

latex_formula_dimension = 'd = 2'

latex_formula_global_optimum = 'f(x1, x2)\\approx-195.629028238419, at$$ $$x1=-0.682584587365898, and$$ $$ x2=-0.36075325513719'

linear = False

modality = False

name = 'Ackley 03'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = True

class opfunu.name_based.a_func.Adjiman(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

[1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark FunctionsFor Global Optimization Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f_{\\text{Adjiman}}(x) = \\cos(x_1)\\sin(x_2) - \\frac{x_1}{(x_2^2 + 1)}'

latex_formula_bounds = 'x_1 \\in [-1.0, 2.0], x_2 \\in [-1., 1.]'

latex_formula_dimension = 'd = 2'

latex_formula_global_optimum = 'f(x1, x2)\\approx-2.02181, at$$ $$x1=2.0, and$$ $$ x2=0.10578'

linear = False

modality = False

name = 'Adjiman Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

class opfunu.name_based.a_func.Alpine01(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

[1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark FunctionsFor Global Optimization Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

continuous = True

convex = True

differentiable = False

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f_{\\text{Alpine01}}(x) = \\sum_{i=1}^{n} \\lvert {x_i \\sin \\left( x_i\\right) + 0.1 x_i} \\rvert'

latex_formula_bounds = 'x_i \\in [-10.0, 10.0], \\forall i \\in \\llbracket 1, d\\rrbracket'

latex_formula_dimension = 'd \\in \\mathbb{N}_{+}^{*}'

latex_formula_global_optimum = 'f(x*)\\approx 0, at$$ $$x*=0.0'

linear = False

modality = False

name = 'Alpine01 Function'

parametric = False

randomized_term = False

scalable = False

separable = True

unimodal = False

class opfunu.name_based.a_func.Alpine02(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

[1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark FunctionsFor Global Optimization Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f_{\\text{Alpine02}(x) = \\prod_{i=1}^{n} \\sqrt{x_i} \\sin(x_i)'

latex_formula_bounds = 'x_i \\in [0., 10.0], \\forall i \\in \\llbracket 1, d\\rrbracket'

latex_formula_dimension = 'd \\in \\mathbb{N}_{+}^{*}'

latex_formula_global_optimum = 'f(x*)\\approx -6.12950, at$$ $$x_1=7.91705268, x_2=4.81584232'

linear = False

modality = False

name = 'Alpine02 Function'

parametric = False

randomized_term = False

scalable = True

separable = True

unimodal = False

opfunu.name_based.b_func module

class opfunu.name_based.b_func.BartelsConn(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

continuous = True

convex = True

differentiable = False

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f_{\\text{BartelsConn}}(x) = \\lvert {x_1^2 + x_2^2 + x_1x_2} \\rvert + \\lvert {\\sin(x_1)} \\rvert + \\lvert {\\cos(x_2)} \\rvert'

latex_formula_bounds = 'x_i \\in [-500, 500], \\forall i \\in \\llbracket 1, d\\rrbracket'

latex_formula_dimension = 'd = 2'

latex_formula_global_optimum = 'f(0, 0) = 1.0'

linear = False

modality = False

name = 'Bartels Conn Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

class opfunu.name_based.b_func.Beale(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

continuous = True

convex = True

differentiable = False

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f_{\\text{Beale}}(x) = \\left(x_1 x_2 - x_1 + 1.5\\right)^{2} +\\left(x_1 x_2^{2} - x_1 + 2.25\right)^{2} + \\left(x_1 x_2^{3} - x_1 + 2.625\right)^{2}'

latex_formula_bounds = 'x_i \\in [-4.5, 4.5], \\forall i \\in \\llbracket 1, d\\rrbracket'

latex_formula_dimension = 'd = 2'

latex_formula_global_optimum = 'f(3.0, 0.5) = 0.0'

linear = False

modality = False

name = 'Beale Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

class opfunu.name_based.b_func.BiggsExp02(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = '\\begin{matrix}f_{\\text{BiggsExp02}}(x) = \\sum_{i=1}^{10} (e^{-t_i x_1} - 5 e^{-t_i x_2} - y_i)^2 \\\\t_i = 0.1 i\\\\y_i = e^{-t_i} - 5 e^{-10t_i}\\\\ \\end{matrix}'

latex_formula_bounds = 'x_i \\in [0., 20.], \\forall i \\in \\llbracket 1, d\\rrbracket'

latex_formula_dimension = 'd = 2'

latex_formula_global_optimum = 'f(1.0, 10.) = 0.0'

linear = False

modality = False

name = 'Biggs EXP2 Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

class opfunu.name_based.b_func.BiggsExp03(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = '\\begin{matrix}\\ f_{\\text{BiggsExp03}}(x) = \\sum_{i=1}^{10} (e^{-t_i x_1} - x_3e^{-t_i x_2} - y_i)^2\\\\t_i = 0.1i\\\\ y_i = e^{-t_i} - 5e^{-10 t_i}\\\\ \\end{matrix}'

latex_formula_bounds = 'x_i \\in [0., 20.], \\forall i \\in \\llbracket 1, d\\rrbracket'

latex_formula_dimension = 'd = 3'

latex_formula_global_optimum = 'f(1.0, 10., 5.0) = 0.0'

linear = False

modality = False

name = 'Biggs EXP3 Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

class opfunu.name_based.b_func.BiggsExp04(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = '\\begin{matrix}\\ f_{\\text{BiggsExp04}}(x) = \\sum_{i=1}^{10} (x_3 e^{-t_i x_1} - x_4 e^{-t_i x_2} - y_i)^2\\\\t_i = 0.1i\\\\ y_i = e^{-t_i} - 5 e^{-10 t_i}\\\\ \\end{matrix}'

latex_formula_bounds = 'x_i \\in [0., 20.], \\forall i \\in \\llbracket 1, d\\rrbracket'

latex_formula_dimension = 'd = 4'

latex_formula_global_optimum = 'f(1.0, 10., 1.0, 5.0) = 0.0'

linear = False

modality = False

name = 'Biggs EXP4 Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

class opfunu.name_based.b_func.BiggsExp05(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = '\\begin{matrix}\\ f_{\\text{BiggsExp05}}(x) = \\sum_{i=1}^{11} (x_3 e^{-t_i x_1} - x_4 e^{-t_i x_2} + 3 e^{-t_i x_5} - y_i)^2\\\\t_i = 0.1i\\\\ y_i = e^{-t_i} - 5e^{-10 t_i} + 3e^{-4 t_i}\\\\ \\end{matrix}'

latex_formula_bounds = 'x_i \\in [0., 20.], \\forall i \\in \\llbracket 1, d\\rrbracket'

latex_formula_dimension = 'd = 5'

latex_formula_global_optimum = 'f(1.0, 10., 1.0, 5.0, 4.0) = 0.0'

linear = False

modality = False

name = 'Biggs EXP5 Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

class opfunu.name_based.b_func.Bird(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f_{\\text{Bird}}(x) = \\left(x_1 - x_2\\right)^{2} + e^{\\left[1 -\\sin\\left(x_1\\right) \\right]^{2}} \\cos\\left(x_2\\right) + e^{\\left[1 - \\cos\\left(x_2\\right)\\right]^{2}} \\sin\\left(x_1\\right)'

latex_formula_bounds = 'x_i \\in [-2\\pi, 2\\pi], \\forall i \\in \\llbracket 1, d\\rrbracket'

latex_formula_dimension = 'd = 2'

latex_formula_global_optimum = 'f(4.701055751981055, 3.152946019601391) = f(-1.582142172055011, -3.130246799635430) = -106.7645367198034'

linear = False

modality = False

name = 'Bird Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

class opfunu.name_based.b_func.Bohachevsky1(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f_{\\text{Bohachevsky}}(x) = \\sum_{i=1}^{n-1}\\left[x_i^2 + 2 x_{i+1}^2 - 0.3 \\cos(3 \\pi x_i) - 0.4 \\cos(4 \\pi x_{i + 1}) + 0.7 \\right]'

latex_formula_bounds = 'x_i \\in [-100, 100], \\forall i \\in \\llbracket 1, d\\rrbracket'

latex_formula_dimension = 'd = 2'

latex_formula_global_optimum = 'f(0, 0) = 0'

linear = False

modality = False

name = 'Bohachevsky 1 Function'

parametric = False

randomized_term = False

scalable = False

separable = True

unimodal = False

class opfunu.name_based.b_func.Bohachevsky2(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f_{\\text{Bohachevsky}}(x) = \\sum_{i=1}^{n-1}\\left[x_i^2 + 2 x_{i+1}^2 - 0.3 \\cos(3 \\pi x_i) - 0.4 \\cos(4 \\pi x_{i + 1}) + 0.7 \\right]'

latex_formula_bounds = 'x_i \\in [-100, 100], \\forall i \\in \\llbracket 1, d\\rrbracket'

latex_formula_dimension = 'd = 2'

latex_formula_global_optimum = 'f(0, 0) = 0'

linear = False

modality = False

name = 'Bohachevsky 2 Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

class opfunu.name_based.b_func.Bohachevsky3(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f_{\\text{Bohachevsky}}(x) = \\sum_{i=1}^{n-1}\\left[x_i^2 + 2 x_{i+1}^2 - 0.3 \\cos(3 \\pi x_i) - 0.4 \\cos(4 \\pi x_{i + 1}) + 0.7 \\right]'

latex_formula_bounds = 'x_i \\in [-100, 100], \\forall i \\in \\llbracket 1, d\\rrbracket'

latex_formula_dimension = 'd = 2'

latex_formula_global_optimum = 'f(0, 0) = 0'

linear = False

modality = False

name = 'Bohachevsky 3 Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

class opfunu.name_based.b_func.Booth(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f_{\\text{Booth}}(x) = (x_1 + 2x_2 - 7)^2 + (2x_1 + x_2 - 5)^2'

latex_formula_bounds = 'x_i \\in [-10, 10], \\forall i \\in \\llbracket 1, d\\rrbracket'

latex_formula_dimension = 'd = 2'

latex_formula_global_optimum = 'f(1, 3) = 0'

linear = False

modality = False

name = 'Booth Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = True

class opfunu.name_based.b_func.BoxBetts(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f_{\\text{BoxBetts}}(x) = \\sum_{i=1}^k g(x_i)^2; g(x) = e^{-0.1i x_1} - e^{-0.1i x_2} - x_3\\left[e^{-0.1i} - e^{-i}\\right]; k=10'

latex_formula_bounds = 'x_1 \\in [0.9, 1.2], x_2 \\in [9, 11.2], x_3 \\in [0.9, 1.2]'

latex_formula_dimension = 'd = 3'

latex_formula_global_optimum = 'f(1, 10, 1) = 0'

linear = False

modality = False

name = 'Box-Betts Quadratic Sum Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

class opfunu.name_based.b_func.Branin01(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f_{\\text{Branin01}}(x) = \\left(- 1.275 \\frac{x_1^{2}}{\\pi^{2}} + 5\\frac{x_1}{\\pi} + x_2 -6\\right)^{2} + \\left(10 -\\frac{5}{4 \\pi} \\right) \\cos\\left(x_1\\right) + 10'

latex_formula_bounds = 'x_1 \\in [-5, 10], x_2 \\in [0, 15]'

latex_formula_dimension = 'd = 2'

latex_formula_global_optimum = 'f(x_i) = 0.39788735772973816, x_i = [-\\pi, 12.275]; or [\\pi, 2.275] or x = [3\\pi, 2.475]'

linear = False

modality = False

name = 'Branin RCOS 1 Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

class opfunu.name_based.b_func.Branin02(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f_{\\text{Branin02}}(x) = \\left(- 1.275 \\frac{x_1^{2}}{\\pi^{2}} + 5 \\frac{x_1}{\\pi} + x_2 - 6 \\right)^{2} + \\left(10 - \\frac{5}{4 \\pi} \\right) \\cos\\left(x_1\\right) \\cos\\left(x_2\\right) + \\log(x_1^2+x_2^2 + 1) + 10'

latex_formula_bounds = 'x_1 \\in [-5, 15], x_2 \\in [-5, 15]'

latex_formula_dimension = 'd = 2'

latex_formula_global_optimum = 'f(x_i) = 5.559037, x_i = [-3.2, 12.53]'

linear = False

modality = False

name = 'Branin RCOS 2 Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

class opfunu.name_based.b_func.Brent(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f_{\\text{Brent}}(x) = (x_1 + 10)^2 + (x_2 + 10)^2 + e^{(-x_1^2 -x_2^2)}'

latex_formula_bounds = 'x_i \\in [-10, 10], \\forall i \\in \\llbracket 1, d\\rrbracket'

latex_formula_dimension = 'd = 2'

latex_formula_global_optimum = 'f(x_i) = 0, x_i = [-10, -10]'

linear = False

modality = False

name = 'Brent Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = True

class opfunu.name_based.b_func.Brown(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f_{\\text{Brown}}(x) = \\sum_{i=1}^{n-1}\\left[\\left(x_i^2\\right)^{x_{i + 1}^2 + 1} + \\left(x_{i + 1}^2\\right)^{x_i^2 + 1}\\right]'

latex_formula_bounds = 'x_i \\in [-1, 4], \\forall i \\in \\llbracket 1, d\\rrbracket'

latex_formula_dimension = 'd = 2'

latex_formula_global_optimum = 'f(x_i) = 0, x_i = 0'

linear = False

modality = False

name = 'Brown Function'

parametric = False

randomized_term = False

scalable = True

separable = False

unimodal = True

class opfunu.name_based.b_func.Bukin02(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f_{\\text{Bukin02}}(x) = 100 (x_2^2 - 0.01x_1^2 + 1) + 0.01(x_1 + 10)^2'

latex_formula_bounds = 'x_1 \\in [-15, -5], x_2 \\in [-3, 3]'

latex_formula_dimension = 'd = 2'

latex_formula_global_optimum = 'f(x_i) = -124.75, x_i = [-15, 0]'

linear = False

modality = False

name = 'Bukin 2 Function'

parametric = False

randomized_term = False

scalable = True

separable = False

unimodal = False

class opfunu.name_based.b_func.Bukin04(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

continuous = True

convex = True

differentiable = False

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f_{\\text{Bukin04}}(x) = 100 x_2^{2} + 0.01 \\lvert{x_1 + 10}\\rvert'

latex_formula_bounds = 'x_1 \\in [-15, -5], x_2 \\in [-3, 3]'

latex_formula_dimension = 'd = 2'

latex_formula_global_optimum = 'f(x_i) = 0, x_i = [-10, 0]'

linear = False

modality = False

name = 'Bukin 4 Function'

parametric = False

randomized_term = False

scalable = True

separable = True

unimodal = False

class opfunu.name_based.b_func.Bukin06(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

continuous = True

convex = True

differentiable = False

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f_{\\text{Bukin06}}(x) = 100 \\sqrt{ \\lvert{x_2 - 0.01 x_1^{2}}\\rvert} + 0.01 \\lvert{x_1 + 10} \\rvert'

latex_formula_bounds = 'x_1 \\in [-15, -5], x_2 \\in [-3, 3]'

latex_formula_dimension = 'd = 2'

latex_formula_global_optimum = 'f(x_i) = 0, x_i = [-10, 1]'

linear = False

modality = False

name = 'Bukin 6 Function'

parametric = False

randomized_term = False

scalable = True

separable = False

unimodal = False

opfunu.name_based.c_func module

class opfunu.name_based.c_func.CamelSixHump(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f(x) = (4 - 2.1x_1^2 + x_1^4/3)x_1^2 + x_1x_2 + (4x_2^2 -4)x_2^2'

latex_formula_bounds = 'x_i \\in [-5, 5], \\forall i \\in \\llbracket 1, d\\rrbracket'

latex_formula_dimension = 'd = 2'

latex_formula_global_optimum = 'f(-0.0898, 0.7126) = f(0.0898, -0.7126) = -1.0316284229280819'

linear = False

modality = False

name = 'Camel Function – Six Hump'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

class opfunu.name_based.c_func.CamelThreeHump(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

continuous = False

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f(x) = 2x_1^2 -1.05x_1^4 + x_1^6/6 + x_1x_2 + x_2^2'

latex_formula_bounds = 'x_i \\in [-5, 5], \\forall i \\in \\llbracket 1, d\\rrbracket'

latex_formula_dimension = 'd = 2'

latex_formula_global_optimum = 'f(0, 0) = 0'

linear = False

modality = False

name = 'Camel Function – Three Hump'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

class opfunu.name_based.c_func.ChenBird(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f(x) = '

latex_formula_bounds = 'x_i \\in [-500, 500], \\forall i \\in \\llbracket 1, d\\rrbracket'

latex_formula_dimension = 'd = 2'

latex_formula_global_optimum = 'f(-113.11622344, 227.73244688) = -1000'

linear = False

modality = False

name = 'Chen Bird Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

class opfunu.name_based.c_func.ChenV(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f(x) = '

latex_formula_bounds = 'x_i \\in [-500, 500], \\forall i \\in \\llbracket 1, d\\rrbracket'

latex_formula_dimension = 'd = 2'

latex_formula_global_optimum = 'f(±0.70710678, ±0.70710678) = -2000.0039999840005'

linear = False

modality = False

name = 'Chen V Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

class opfunu.name_based.c_func.Chichinadze(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f(x) = x_{1}^{2} - 12 x_{1} + 8 \\sin\\left(\\frac{5}{2} \\pi x_{1}\\right)+ 10 \\cos\\left(\\frac{1}{2} \\pi x_{1}\\right) + 11 - 0.2 \\frac{\\sqrt{5}}{e^{\\frac{1}{2} \\left(x_{2} -0.5 \\right)^{2}}}'

latex_formula_bounds = 'x_i \\in [-30, 30], \\forall i \\in \\llbracket 1, d\\rrbracket'

latex_formula_dimension = 'd = 2'

latex_formula_global_optimum = 'f(6.189866586965680, 0.5) = -42.94438701899098'

linear = False

modality = False

name = 'Chichinadze Function'

parametric = False

randomized_term = False

scalable = False

separable = True

unimodal = False

class opfunu.name_based.c_func.ChungReynolds(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f(x) = (\\sum_{i=1}^D x_i^2)^2'

latex_formula_bounds = 'x_i \\in [-100, 100], \\forall i \\in \\llbracket 1, d\\rrbracket'

latex_formula_dimension = 'd \\in \\mathbb{N}_{+}^{*}'

latex_formula_global_optimum = 'f(0,...,0) = 0'

linear = False

modality = False

name = 'Chung Reynolds Function'

parametric = False

randomized_term = False

scalable = True

separable = False

unimodal = True

class opfunu.name_based.c_func.Cigar(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Gavana, A. Global Optimization Benchmarks and AMPGO retrieved 2022

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f(x) = x_1^2 + 10^6\\sum_{i=2}^{n} x_i^2'

latex_formula_bounds = 'x_i \\in [-100, 100], \\forall i \\in \\llbracket 1, d\\rrbracket'

latex_formula_dimension = 'd \\in \\mathbb{N}_{+}^{*}'

latex_formula_global_optimum = 'f(0,...,0) = 0'

linear = False

modality = False

name = 'Cigar Function'

parametric = False

randomized_term = False

scalable = True

separable = True

unimodal = True

class opfunu.name_based.c_func.Cola(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f(x) = x_1^2 + 10^6\\sum_{i=2}^{n} x_i^2'

latex_formula_bounds = 'x_0 \\in [0, 4], x_i \\in [-4, 4], \\forall i \\in \\llbracket 1, d-1\\rrbracket'

latex_formula_dimension = 'd = 17'

latex_formula_global_optimum = 'f(0,...,0) = 11.7464'

linear = False

modality = True

name = 'Cola Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

class opfunu.name_based.c_func.Colville(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f(x) = \\left(x_{1} -1\\right)^{2} + 100 \\left(x_{1}^{2} - x_{2}\\right)^{2} + 10.1 \\left(x_{2} -1\\right)^{2} + \\left(x_{3} -1\\right)^{2} + 90 \\left(x_{3}^{2} - x_{4}\\right)^{2} + 10.1 \\left(x_{4} -1\\right)^{2} + 19.8 \\frac{x_{4} -1}{x_{2}}'

latex_formula_bounds = 'x_i \\in [-10, 10], \\forall i \\in \\llbracket 1, d\\rrbracket'

latex_formula_dimension = 'd = 4'

latex_formula_global_optimum = 'f(1,...,1) = 0'

linear = False

modality = False

name = 'Colville Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

class opfunu.name_based.c_func.Corana(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

continuous = False

convex = False

differentiable = False

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f(x) = '

latex_formula_bounds = 'x_i \\in [-5, 5], \\forall i \\in \\llbracket 1, d\\rrbracket'

latex_formula_dimension = 'd = 4'

latex_formula_global_optimum = 'f(1,...,1) = 0'

linear = False

modality = False

name = 'Corana Function'

parametric = False

randomized_term = False

scalable = False

separable = True

unimodal = False

class opfunu.name_based.c_func.CosineMixture(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

continuous = False

convex = False

differentiable = False

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f(x) = -0.1 \\sum_{i=1}^n \\cos(5 \\pi x_i) - \\sum_{i=1}^n x_i^2'

latex_formula_bounds = 'x_i \\in [-1, 1], \\forall i \\in \\llbracket 1, d\\rrbracket'

latex_formula_dimension = 'd = 2'

latex_formula_global_optimum = 'f(-1,...,-1) = -0.9*D'

linear = False

modality = False

name = 'Cosine Mixture Function'

parametric = False

randomized_term = False

scalable = False

separable = True

unimodal = False

class opfunu.name_based.c_func.CrossInTray(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

continuous = True

convex = False

differentiable = False

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f(x) = - 0.0001 \\left(\\left|{e^{\\left|{100- \\frac{\\sqrt{x_{1}^{2} + x_{2}^{2}}}{\\pi}}\\right|} \\sin\\left(x_{1}\\right) \\sin\\left(x_{2}\\right)}\\right| + 1\\right)^{0.1}'

latex_formula_bounds = 'x_i \\in [-10, 10], \\forall i \\in \\llbracket 1, d\\rrbracket'

latex_formula_dimension = 'd = 2'

latex_formula_global_optimum = 'f(±1.349406608602084, ±1.349406608602084) = -2.062611870822739'

linear = False

modality = False

name = 'Cross-in-Tray Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

class opfunu.name_based.c_func.CrossLegTable(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Mishra, S. Global Optimization by Differential Evolution and Particle Swarm Methods:

Evaluation on Some Benchmark Functions Munich University, 2006

continuous = True

convex = False

differentiable = False

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f(x) = -\\frac{1}{\\left(\\left|{e^{\\left|{100 - \\frac{\\sqrt{x_{1}^{2} + x_{2}^{2}}}{\\pi}}\\right|}\\sin\\left(x_{1}\\right) \\sin\\left(x_{2}\\right)}\\right| + 1\\right)^{0.1}}'

latex_formula_bounds = 'x_i \\in [-10, 10], \\forall i \\in \\llbracket 1, d\\rrbracket'

latex_formula_dimension = 'd = 2'

latex_formula_global_optimum = 'f(0, 0) = -1'

linear = False

modality = False

name = 'Cross-Leg-Table Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

class opfunu.name_based.c_func.CrownedCross(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Mishra, S. Global Optimization by Differential Evolution and Particle Swarm Methods:

Evaluation on Some Benchmark Functions Munich University, 2006

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f(x) = 0.0001 \\left(\\left|{e^{\\left|{100 - \\frac{\\sqrt{x_{1}^{2} + x_{2}^{2}}}{\\pi}}\\right|}\\sin\\left(x_{1}\\right) \\sin\\left(x_{2}\\right)}\\right| + 1\\right)^{0.1}'

latex_formula_bounds = 'x_i \\in [-10, 10], \\forall i \\in \\llbracket 1, d\\rrbracket'

latex_formula_dimension = 'd = 2'

latex_formula_global_optimum = 'f(0, 0) = 0.0001'

linear = False

modality = False

name = 'Cross-Leg-Table Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

class opfunu.name_based.c_func.Csendes(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Mishra, S. Global Optimization by Differential Evolution and Particle Swarm Methods:

Evaluation on Some Benchmark Functions Munich University, 2006

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f(x) = \\sum_{i=1}^n x_i^6 \\left[ 2 + \\sin \\left( \\frac{1}{x_i} \\right ) \\right]'

latex_formula_bounds = 'x_i \\in [-1, 1], \\forall i \\in \\llbracket 1, d\\rrbracket'

latex_formula_dimension = 'd \\in \\mathbb{N}_{+}^{*}'

latex_formula_global_optimum = 'f(0,..., 0) = 0'

linear = False

modality = False

name = 'Csendes Function'

parametric = False

randomized_term = False

scalable = True

separable = True

unimodal = False

class opfunu.name_based.c_func.Cube(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Mishra, S. Global Optimization by Differential Evolution and Particle Swarm Methods:

Evaluation on Some Benchmark Functions Munich University, 2006

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f(x) = 100(x_2 - x_1^3)^2 + (1 - x1)^2'

latex_formula_bounds = 'x_i \\in [-10, 10], \\forall i \\in \\llbracket 1, d\\rrbracket'

latex_formula_dimension = 'd \\in \\mathbb{N}_{+}^{*}'

latex_formula_global_optimum = 'f(1, 1) = 0'

linear = False

modality = False

name = 'Cube Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

opfunu.name_based.d_func module

class opfunu.name_based.d_func.Damavandi(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f(x) = \\left[ 1 - \\lvert{\\frac{\\sin[\\pi (x_1 - 2)]\\sin[\\pi (x2 - 2)]}{\\pi^2 (x_1 - 2)(x_2 - 2)}}\\rvert^5 \\right] \\left[2 + (x_1 - 7)^2 + 2(x_2 - 7)^2 \\right]'

latex_formula_bounds = 'x_i \\in [0, 14], \\forall i \\in \\llbracket 1, d\\rrbracket'

latex_formula_dimension = 'd = 2'

latex_formula_global_optimum = 'f(2, 2) = 0'

linear = False

modality = False

name = 'Damavandi Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

class opfunu.name_based.d_func.DeVilliersGlasser01(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f(x) = \\sum_{i=1}^{24} \\left[ x_1x_2^{t_i}\\sin(x_3t_i + x_4) - y_i \\right ]^2'

latex_formula_bounds = 'x_i \\in [-500, 500], \\forall i \\in \\llbracket 1, d\\rrbracket'

latex_formula_dimension = 'd = 4'

latex_formula_global_optimum = 'f(60.137, 1.371, 3.112, 1.761) = 0'

linear = False

modality = True

name = 'DeVilliers-Glasser 1 Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

class opfunu.name_based.d_func.DeVilliersGlasser02(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f(x) = \\sum_{i=1}^{24} \\left[ x_1x_2^{t_i}\\tanh \\left [x_3t_i + \\sin(x_4t_i) \\right] \\cos(t_ie^{x_5}) - y_i \\right ]^2'

latex_formula_bounds = 'x_i \\in [-500, 500], \\forall i \\in \\llbracket 1, d\\rrbracket'

latex_formula_dimension = 'd = 5'

latex_formula_global_optimum = 'f(53.81, 1.27, 3.012, 2.13, 0.507) = 0'

linear = False

modality = False

name = 'DeVilliers-Glasser 2 Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

class opfunu.name_based.d_func.Deb01(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f(x) = - \\frac{1}{N} \\sum_{i=1}^n \\sin^6(5 \\pi x_i)'

latex_formula_bounds = 'x_i \\in [-1, 1], \\forall i \\in \\llbracket 1, d\\rrbracket'

latex_formula_dimension = 'd = 2'

latex_formula_global_optimum = 'f(0.3, -0.3) = -1'

linear = False

modality = True

name = 'Deb 1 Function'

parametric = False

randomized_term = False

scalable = False

separable = True

unimodal = False

class opfunu.name_based.d_func.Deb03(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f(x) = - \\frac{1}{N} \\sum_{i=1}^n \\sin^6 \\left[ 5 \\pi\\left ( x_i^{3/4} - 0.05 \\right) \\right ]'

latex_formula_bounds = 'x_i \\in [-1, 1], \\forall i \\in \\llbracket 1, d\\rrbracket'

latex_formula_dimension = 'd = 2'

latex_formula_global_optimum = 'f(0.3, -0.3) = -1'

linear = False

modality = True

name = 'Deb 3 Function'

parametric = False

randomized_term = False

scalable = False

separable = True

unimodal = False

class opfunu.name_based.d_func.Decanomial(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Gavana, A. Global Optimization Benchmarks and AMPGO retrieved 2015

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f(x) = 0.001 \\left(\\lvert{x_{2}^{4} + 12 x_{2}^{3}+ 54 x_{2}^{2} + 108 x_{2} + 81.0}\\rvert + \\lvert{x_{1}^{10} - 20 x_{1}^{9} + 180 x_{1}^{8} - 960 x_{1}^{7} + 3360 x_{1}^{6}- 8064 x_{1}^{5} + 13340 x_{1}^{4} - 15360 x_{1}^{3} + 11520 x_{1}^{2} - 5120 x_{1} + 2624.0}\\rvert\\right)^{2}'

latex_formula_bounds = 'x_i \\in [-10, 10], \\forall i \\in \\llbracket 1, d\\rrbracket'

latex_formula_dimension = 'd = 2'

latex_formula_global_optimum = 'f(2, -3) = 0'

linear = False

modality = False

name = 'Decanomial Function'

parametric = False

randomized_term = False

scalable = False

separable = True

unimodal = False

class opfunu.name_based.d_func.Deceptive(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Gavana, A. Global Optimization Benchmarks and AMPGO retrieved 2015

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f(x) = - \\left [\\frac{1}{n} \\sum_{i=1}^{n} g_i(x_i) \\right ]^{\\beta}'

latex_formula_bounds = 'x_i \\in [0, 1], \\forall i \\in \\llbracket 1, d\\rrbracket'

latex_formula_dimension = 'd \\in N^+'

latex_formula_global_optimum = 'f(alpha_i) = -1'

linear = False

modality = False

name = 'Deceptive Function'

parametric = False

randomized_term = False

scalable = True

separable = False

unimodal = False

class opfunu.name_based.d_func.DeckkersAarts(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f(x) = 10^5x_1^2 + x_2^2 - (x_1^2 + x_2^2)^2 + 10^{-5}(x_1^2 + x_2^2)^4'

latex_formula_bounds = 'x_i \\in [-20, 20], \\forall i \\in \\llbracket 1, d\\rrbracket'

latex_formula_dimension = 'd = 2'

latex_formula_global_optimum = 'f(0, \\pm 14.9451209) = -24776.518242168'

linear = False

modality = False

name = 'Deckkers-Aarts Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

class opfunu.name_based.d_func.DeflectedCorrugatedSpring(ndim=None, bounds=None, alpha=5.0)

Bases: opfunu.benchmark.Benchmark

1

Gavana, A. Global Optimization Benchmarks and AMPGO retrieved 2015

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f(x) = 0.1\\sum_{i=1}^n \\left[ (x_i - \\alpha)^2 - \\cos \\left( K \\sqrt {\\sum_{i=1}^n (x_i - \\alpha)^2}\\right ) \\right ]'

latex_formula_bounds = 'x_i \\in [-0, 2\\alpha], \\forall i \\in \\llbracket 1, d\\rrbracket'

latex_formula_dimension = 'd = 2'

latex_formula_global_optimum = 'f(x_i) = f(alpha_i) = -1'

linear = False

modality = True

name = 'Deflected Corrugated Spring Function'

parametric = False

randomized_term = False

scalable = True

separable = False

unimodal = False

class opfunu.name_based.d_func.DixonPrice(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f(x) = (x_i - 1)^2 + \\sum_{i=2}^n i(2x_i^2 - x_{i-1})^2'

latex_formula_bounds = 'x_i \\in [-10, 10], \\forall i \\in \\llbracket 1, d\\rrbracket'

latex_formula_dimension = 'd \\in N^+'

latex_formula_global_optimum = 'f(2^{- \\frac{(2^i - 2)}{2^i}}) = 0'

linear = False

modality = False

name = 'Dixon & Price Function'

parametric = False

randomized_term = False

scalable = True

separable = False

unimodal = True

class opfunu.name_based.d_func.Dolan(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f(x) = \\lvert (x_1 + 1.7 x_2)\\sin(x_1) - 1.5 x_3 - 0.1 x_4\\cos(x_5 + x_5 - x_1) + 0.2 x_5^2 - x_2 - 1 \\rvert'

latex_formula_bounds = 'x_i \\in [-100, 100], \\forall i \\in \\llbracket 1, d\\rrbracket'

latex_formula_dimension = 'd = 5'

latex_formula_global_optimum = 'f(8.39045925, 4.81424707, 7.34574133, 68.88246895, 3.85470806) = 0'

linear = False

modality = False

name = 'Dolan Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

class opfunu.name_based.d_func.DropWave(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Gavana, A. Global Optimization Benchmarks and AMPGO retrieved 2015

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f(x) = - \\frac{1 + \\cos\\left(12 \\sqrt{\\sum_{i=1}^{n} x_i^{2}}\\right)}{2 + 0.5 \\sum_{i=1}^{n} x_i^{2}}'

latex_formula_bounds = 'x_i \\in [-5.12, 5.12], \\forall i \\in \\llbracket 1, d\\rrbracket'

latex_formula_dimension = 'd = 2'

latex_formula_global_optimum = 'f(0, 0) = -1'

linear = False

modality = False

name = 'DropWave Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

opfunu.name_based.e_func module

class opfunu.name_based.e_func.Easom(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f(x) = a - \\frac{a}{e^{b \\sqrt{\\frac{\\sum_{i=1}^{n}x_i^{2}}{n}}}} + e - e^{\\frac{\\sum_{i=1}^{n} \\cos\\left(c x_i\\right)} {n}}'

latex_formula_bounds = 'x_i \\in [-100, 100], \\forall i \\in \\llbracket 1, d\\rrbracket'

latex_formula_dimension = 'd = 2'

latex_formula_global_optimum = 'f(pi, pi) = -1'

linear = False

modality = False

name = 'Easom Function'

parametric = False

randomized_term = False

scalable = False

separable = True

unimodal = False

class opfunu.name_based.e_func.Eckerle4(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

[1] Eckerle, K., NIST (1979). Circular Interference Transmittance Study. [2] https://www.itl.nist.gov/div898/strd/nls/data/eckerle4.shtml

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f(x) = '

latex_formula_bounds = '0 <= x_1 <=20, 1 <= x_2 <= 20, 10 <= x_3 <= 600'

latex_formula_dimension = 'd = 3'

latex_formula_global_optimum = 'f(1.5543827178, 4.0888321754, 4.5154121844e2) = 1.4635887487E-03'

linear = False

modality = False

name = 'Eckerle 4 Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

class opfunu.name_based.e_func.EggCrate(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f(x) = x_1^2 + x_2^2 + 25 \\left[ \\sin^2(x_1) + \\sin^2(x_2) \\right]'

latex_formula_bounds = 'x_i \\in [-5, 5], \\forall i \\in \\llbracket 1, d\\rrbracket'

latex_formula_dimension = 'd = 2'

latex_formula_global_optimum = 'f(0, 0) = 0'

linear = False

modality = False

name = 'Egg Crate Function'

parametric = False

randomized_term = False

scalable = False

separable = True

unimodal = False

class opfunu.name_based.e_func.EggHolder(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f(x) = \\sum_{1}^{n - 1}\\left[-\\left(x_{i + 1}+ 47 \\right ) \\sin\\sqrt{\\lvert x_{i+1} + x_i/2 + 47 \\rvert} - x_i \\sin\\sqrt{\\lvert x_i - (x_{i + 1} + 47)\\rvert}\\right ]'

latex_formula_bounds = 'x_i \\in [-512, 512], \\forall i \\in \\llbracket 1, d\\rrbracket'

latex_formula_dimension = 'd \\in N^+'

latex_formula_global_optimum = 'f(512, 404.2319) = -959.640662711'

linear = False

modality = False

name = 'Egg Holder Function'

parametric = False

randomized_term = False

scalable = True

separable = False

unimodal = False

class opfunu.name_based.e_func.ElAttarVidyasagarDutta(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f(x) = (x_1^2 + x_2 - 10)^2 + (x_1 + x_2^2 - 7)^2 + (x_1^2 + x_2^3 - 1)^2'

latex_formula_bounds = 'x_i \\in [-500, 500], \\forall i \\in \\llbracket 1, d\\rrbracket'

latex_formula_dimension = 'd = 2'

latex_formula_global_optimum = 'f(3.40918683, -2.17143304) = 1.712780354'

linear = False

modality = False

name = 'El-Attar-Vidyasagar-Dutta Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = True

class opfunu.name_based.e_func.Exp2(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f(x) = \\sum_{i=0}^9 \\left ( e^{-ix_1/10} - 5e^{-ix_2/10} - e^{-i/10} + 5e^{-i} \\right )^2'

latex_formula_bounds = 'x_i \\in [0, 20], \\forall i \\in \\llbracket 1, d\\rrbracket'

latex_formula_dimension = 'd = 2'

latex_formula_global_optimum = 'f(1, 10) = 0'

linear = False

modality = False

name = 'Exp 2 Function'

parametric = False

randomized_term = False

scalable = False

separable = True

unimodal = False

class opfunu.name_based.e_func.Exponential(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f(x) = -e^{-0.5 \\sum_{i=1}^n x_i^2}'

latex_formula_bounds = 'x_i \\in [-1, 1], \\forall i \\in \\llbracket 1, d\\rrbracket'

latex_formula_dimension = 'd \\in N^+'

latex_formula_global_optimum = 'f(0,..,0) = -1'

linear = False

modality = False

name = 'Exponential Function'

parametric = False

randomized_term = False

scalable = True

separable = False

unimodal = False

opfunu.name_based.f_func module

class opfunu.name_based.f_func.FreudensteinRoth(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f(x) = \\left\\{x_1 - 13 + \\left[(5 - x_2) x_2- 2 \\right] x_2 \\right\\}^2 + \\left \\{x_1 - 29 + \\left[(x_2 + 1) x_2 - 14 \\right] x_2 \\right\\}^2'

latex_formula_bounds = 'x_i \\in [-10, 10], \\forall i \\in \\llbracket 1, d\\rrbracket'

latex_formula_dimension = 'd = 2'

latex_formula_global_optimum = 'f(5, 4) = 0'

linear = False

modality = False

name = 'Freudenstein Roth Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

opfunu.name_based.g_func module

class opfunu.name_based.g_func.Gear(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Gavana, A. Global Optimization Benchmarks and AMPGO retrieved 2015

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f(x) = \\left \\{ \\frac{1.0}{6.931}- \\frac{\\lfloor x_1\\rfloor \\lfloor x_2 \\rfloor } {\\lfloor x_3 \\rfloor \\lfloor x_4 \\rfloor } \\right\\}^2'

latex_formula_bounds = 'x_i \\in [12, 60], \\forall i \\in \\llbracket 1, d\\rrbracket'

latex_formula_dimension = 'd = 4'

latex_formula_global_optimum = 'f(16, 19, 43, 49) = 2.7 \\cdot 10^{-12}'

linear = False

modality = False

name = 'Gear Problem'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

class opfunu.name_based.g_func.Giunta(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f(x) = 0.6 + \\sum_{i=1}^{n} \\left[\\sin^{2}\\left(1- \\frac{16}{15} x_i\\right) - \\frac{1}{50} \\sin\\left(4 - \\frac{64}{15} x_i\\right) - \\sin\\left(1 - \\frac{16}{15} x_i\\right)\\right]'

latex_formula_bounds = 'x_i \\in [-1, 1], \\forall i \\in \\llbracket 1, d\\rrbracket'

latex_formula_dimension = 'd = 2'

latex_formula_global_optimum = 'f([0.4673200277395354, 0.4673200169591304]) = 0.06447042053690566'

linear = False

modality = False

name = 'Giunta Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

class opfunu.name_based.g_func.GoldsteinPrice(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f(x) = \\left[ 1 + (x_1 + x_2 + 1)^2 (19 - 14 x_1 + 3 x_1^2 - 14 x_2 + 6 x_1 x_2 + 3 x_2^2) \\right] \\left[ 30 + ( 2x_1 - 3 x_2)^2 (18 - 32 x_1 + 12 x_1^2 + 48 x_2 - 36 x_1 x_2 + 27 x_2^2) \\right]'

latex_formula_bounds = 'x_i \\in [-2, 2], \\forall i \\in \\llbracket 1, d\\rrbracket'

latex_formula_dimension = 'd = 2'

latex_formula_global_optimum = 'f([0, -1]) = 3'

linear = False

modality = False

name = 'Goldstein Price Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

class opfunu.name_based.g_func.Griewank(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f(x) = \\frac{1}{4000}\\sum_{i=1}^n x_i^2 - \\prod_{i=1}^n\\cos\\left(\\frac{x_i}{\\sqrt{i}}\\right) + 1'

latex_formula_bounds = 'x_i \\in [-100, 100], \\forall i \\in \\llbracket 1, d\\rrbracket'

latex_formula_dimension = 'd \\in N^+'

latex_formula_global_optimum = 'f(0,...,0) = 0'

linear = False

modality = False

name = 'Griewank Function'

parametric = False

randomized_term = False

scalable = True

separable = False

unimodal = False

class opfunu.name_based.g_func.Gulf(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f(x) = \\sum_{i=1}^99 \\left( e^{-\\frac{\\lvert y_i - x_2 \\rvert^{x_3}}{x_1}}  - t_i \\right)'

latex_formula_bounds = 'x_i \\in [0, 60], \\forall i \\in \\llbracket 1, d\\rrbracket'

latex_formula_dimension = 'd = 3'

latex_formula_global_optimum = 'f(50, 25, 1.5) = 0'

linear = False

modality = False

name = 'Gulf Research Problem'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

opfunu.name_based.h_func module

class opfunu.name_based.h_func.Hansen(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f(x) = \\left[ \\sum_{i=0}^4(i+1)\\cos(ix_1+i+1)\\right ]\\left[\\sum_{j=0}^4(j+1)\\cos[(j+2)x_2+j+1])\\right ]'

latex_formula_bounds = 'x_i \\in [-10, 10], \\forall i \\in \\llbracket 1, d\\rrbracket'

latex_formula_dimension = 'd = 2'

latex_formula_global_optimum = 'f(-7.58989583, -7.70831466) = -176.54179'

linear = False

modality = False

name = 'Hansen Function'

parametric = False

randomized_term = False

scalable = False

separable = True

unimodal = False

class opfunu.name_based.h_func.Hartmann3(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f(x) = -\\sum\\limits_{i=1}^{4} c_i e^{-\\sum\\limits_{j=1}^{n}a_{ij}(x_j - p_{ij})^2}'

latex_formula_bounds = 'x_i \\in [0, 1], \\forall i \\in \\llbracket 1, d\\rrbracket'

latex_formula_dimension = 'd = 3'

latex_formula_global_optimum = 'f([0.11461292,  0.55564907,  0.85254697]) = -3.8627821478'

linear = False

modality = False

name = 'Hartman 3 Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

class opfunu.name_based.h_func.Hartmann6(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f(x) = -\\sum\\limits_{i=1}^{4} c_i e^{-\\sum\\limits_{j=1}^{n}a_{ij}(x_j - p_{ij})^2}'

latex_formula_bounds = 'x_i \\in [0, 1], \\forall i \\in \\llbracket 1, d\\rrbracket'

latex_formula_dimension = 'd = 3'

latex_formula_global_optimum = 'f([0.20168952, 0.15001069, 0.47687398, 0.27533243, 0.31165162, 0.65730054]) = -3.32236801141551'

linear = False

modality = False

name = 'Hartman 6 Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

class opfunu.name_based.h_func.HelicalValley(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f(x) = 100{[z-10\\Psi(x_1,x_2)]^2 +(\\sqrt{x_1^2+x_2^2}-1)^2}+x_3^2'

latex_formula_bounds = 'x_i \\in [-10, 10], \\forall i \\in \\llbracket 1, d\\rrbracket'

latex_formula_dimension = 'd \\in N^+'

latex_formula_global_optimum = 'f([1.0, 0.0, 0.0]) = 0'

linear = False

modality = False

name = 'Helical Valley'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

class opfunu.name_based.h_func.Himmelblau(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f(x) = (x_1^2 + x_2 - 11)^2 + (x_1 + x_2^2 - 7)^2'

latex_formula_bounds = 'x_i \\in [-5, 5], \\forall i \\in \\llbracket 1, d\\rrbracket'

latex_formula_dimension = 'd \\in N^+'

latex_formula_global_optimum = 'f([3, 2]) = 0'

linear = False

modality = False

name = 'Himmelblau Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

class opfunu.name_based.h_func.HolderTable(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Gavana, A. Global Optimization Benchmarks and AMPGO retrieved 2015

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f(x) = - \\left|{e^{\\left|{1- \\frac{\\sqrt{x_{1}^{2} + x_{2}^{2}}}{\\pi} }\\right|} \\sin\\left(x_{1}\\right) \\cos\\left(x_{2}\\right)}\\right|'

latex_formula_bounds = ' 0 <= x_1 <= 5, 0 <= x2 <= 6'

latex_formula_dimension = 'd = 2'

latex_formula_global_optimum = 'f(\\pm 9.664590028909654) = -19.20850256788675'

linear = False

modality = False

name = 'Hosaki Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

class opfunu.name_based.h_func.Hosaki(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f(x) = \\left ( 1 - 8 x_1 + 7 x_1^2 - \\frac{7}{3} x_1^3 + \\frac{1}{4} x_1^4 \\right ) x_2^2 e^{-x_1}'

latex_formula_bounds = ' 0 <= x_1 <= 5, 0 <= x2 <= 6'

latex_formula_dimension = 'd = 2'

latex_formula_global_optimum = 'f(4, 2) = −2.3458'

linear = False

modality = False

name = 'Hosaki Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

opfunu.name_based.i_func module

class opfunu.name_based.i_func.Infinity(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f(x) = \\sum_{i=1}^{n} x_i^{6} \\left [ \\sin\\left ( \\frac{1}{x_i} \\right ) + 2 \\right ]'

latex_formula_bounds = 'x_i \\in [-1, 1], \\forall i \\in \\llbracket 1, d\\rrbracket'

latex_formula_dimension = 'd \\in N^+'

latex_formula_global_optimum = 'f(0,..,0) = 0'

linear = False

modality = False

name = 'Hansen Function'

parametric = False

randomized_term = False

scalable = True

separable = True

unimodal = False

opfunu.name_based.j_func module

class opfunu.name_based.j_func.JennrichSampson(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f_{\\text{JennrichSampson}}(x) = \\sum_{i=1}^{10} \\left [2 + 2i - (e^{ix_1} + e^{ix_2}) \\right ]^2'

latex_formula_bounds = 'x_i \\in [-1, 1]'

latex_formula_dimension = 'd = 2'

latex_formula_global_optimum = 'f(0.257825, 0.257825) = 124.3621824'

linear = False

modality = False

name = 'Jennrich-Sampson Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

class opfunu.name_based.j_func.Judge(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Gavana, A. Global Optimization Benchmarks and AMPGO retrieved 2015

\[f_{ ext{Judge}}(x) = \sum_{i=1}^{20} \left [ \left (x_1 + A_i x_2 + B x_2^2\]

ight ) - C_i ight ]^2

\[egin{cases} C = [4.284, 4.149, 3.877, 0.533, 2.211, 2.389, 2.145, 3.231, 1.998, 1.379, 2.106, 1.428, 1.011, 2.179, 2.858, 1.388, 1.651, 1.593, 1.046, 2.152] \ A = [0.286, 0.973, 0.384, 0.276, 0.973, 0.543, 0.957, 0.948, 0.543, 0.797, 0.936, 0.889, 0.006, 0.828, 0.399, 0.617, 0.939, 0.784, 0.072, 0.889] \ B = [0.645, 0.585, 0.310, 0.058, 0.455, 0.779, 0.259, 0.202, 0.028, 0.099, 0.142, 0.296, 0.175, 0.180, 0.842, 0.039, 0.103, 0.620, 0.158, 0.704] \end{cases}\]

with \(x_i \in [-10, 10]\) for \(i = 1, 2\). Global optimum: \(f(x_i) = 16.0817307\) for \(\mathbf{x} = [0.86479, 1.2357]\).

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f_{\\text{Judge}}(x) = \\sum_{i=1}^{20} \\left [ \\left (x_1 + A_i x_2 + B x_2^2 \\right ) - C_i \\right ]^2'

latex_formula_bounds = 'x_i \\in [-10, 10]'

latex_formula_dimension = 'd \\in N^+'

latex_formula_global_optimum = 'f(0.86479, 1.2357) = 16.0817307'

linear = False

modality = False

name = 'Judge Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

opfunu.name_based.k_func module

class opfunu.name_based.k_func.Katsuura(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Adorio, E. MVF - “Multivariate Test Functions Library in C for

Unconstrained Global Optimization”, 2005 [2] Gavana, A. Global Optimization Benchmarks and AMPGO retrieved 2015

\[f_{ ext{Katsuura}}(x) = \prod_{i=0}^{n-1} \left [ 1 + (i+1) \sum_{k=1}^{d} \lfloor (2^k x_i)\]

floor 2^{-k} ight ]

Where, in this exercise, \(d = 32\). Here, \(n\) represents the number of dimensions and

\(x_i \in [0, 100]\) for \(i = 1, ..., n\).

Global optimum: \(f(x) = 1\) for \(x_i = 0\) for \(i = 1, ..., n\).

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f_{\\text{Katsuura}}(x) = \\prod_{i=0}^{n-1} \\left [ 1 + (i+1) \\sum_{k=1}^{d} \\lfloor (2^k x_i) \\rfloor 2^{-k} \\right ]'

latex_formula_bounds = 'x_i \\in [0, 100]'

latex_formula_dimension = 'd = 32'

latex_formula_global_optimum = 'f(0., 0., ..., 0.) = 1.'

linear = False

modality = False

name = 'Katsuura Function'

parametric = False

randomized_term = False

scalable = True

separable = False

unimodal = False

class opfunu.name_based.k_func.Keane(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

\[f_{ ext{Keane}}(x) =\]

rac{sin^2(x_1 - x_2)sin^2(x_1 + x_2)}{sqrt{x_1^2 + x_2^2}}

with \(x_i \in [0, 10]\) for \(i = 1, 2\).

Global optimum: \(f(x) = 0.0\) for \(x = [7.85396153, 7.85396135]\).

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f_{\\text{Keane}}(x) = \\frac{\\sin^2(x_1 - x_2)\\sin^2(x_1 + x_2)} {\\sqrt{x_1^2 + x_2^2}}'

latex_formula_bounds = 'x_i \\in [0, 10]'

latex_formula_dimension = 'd = 2'

latex_formula_global_optimum = 'f(7.85396153, 7.85396135) = 0.'

linear = False

modality = True

name = 'Katsuura Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

class opfunu.name_based.k_func.Kowalik(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

\[f_{ ext{Kowalik}}(x) = \sum_{i=0}^{10} \left [ a_i -\]

rac{x_1 (b_i^2 + b_i x_2)} {b_i^2 + b_i x_3 + x_4} ight ]^2

\[egin{matrix} a = [4, 2, 1, 1/2, 1/4 1/8, 1/10, 1/12, 1/14, 1/16] \ b = [0.1957, 0.1947, 0.1735, 0.1600, 0.0844, 0.0627, 0.0456, 0.0342, 0.0323, 0.0235, 0.0246]\ \end{matrix}\]

Here, \(n\) represents the number of dimensions and \(x_i \in [-5, 5]\) for \(i = 1, ..., 4\).

Global optimum: \(f(x) = 0.00030748610\) for \(x = [0.192833, 0.190836, 0.123117, 0.135766]\).

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f_{\\text{Kowalik}}(x) = \\sum_{i=0}^{10} \\left [ a_i - \\frac{x_1 (b_i^2 + b_i x_2)} {b_i^2 + b_i x_3 + x_4} \\right ]^2'

latex_formula_bounds = 'x_i \\in [-5, 5]'

latex_formula_dimension = 'd = 4'

latex_formula_global_optimum = 'f(0.192833, 0.190836, 0.123117, 0.135766) = 0.00030748610'

linear = False

modality = False

name = 'Kowalik Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

opfunu.name_based.l_func module

class opfunu.name_based.l_func.Langermann(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Gavana, A. Global Optimization Benchmarks and AMPGO retrieved 2015

\[f_{ ext{Langermann}}(x) = - \sum_{i=1}^{5}\]

rac{c_i cosleft{pi left[left(x_{1}- a_i ight)^{2}

• left(x_{2} - b_i

ight)^{2} ight] ight}}{e^{ rac{left( x_{1}

• a_i

ight)^{2} + left( x_{2} - b_i ight)^{2}}{pi}}}

Where: .. math:

egin{matrix}
a = [3, 5, 2, 1, 7]\
b = [5, 2, 1, 4, 9]\
c = [1, 2, 5, 2, 3] \
\end{matrix}

Here \(x_i \in [0, 10]\) for \(i = 1, 2\). Global optimum: \(f(x) = -5.1621259\)

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f_{\\text{Langermann}}(x) = - \\sum_{i=1}^{5} \\frac{c_i \\cos\\left\\{\\pi \\left[\\left(x_{1}- a_i\\right)^{2}  + \\left(x_{2} - b_i \\right)^{2}\\right]\\right\\}}{e^{\\frac{\\left( x_{1} - a_i\\right)^{2} + \\left( x_{2} - b_i\\right)^{2}}{\\pi}}}'

latex_formula_bounds = 'x_i \\in [0, 10]'

latex_formula_dimension = 'd = 2'

latex_formula_global_optimum = 'f(2.00299219, 1.006096) = -5.1621259'

linear = False

modality = True

name = 'Langermann Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

class opfunu.name_based.l_func.LennardJones(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

http://www-wales.ch.cam.ac.uk/~jon/structures/LJ/tables.150.html

\[f_{ ext{LennardJones}}(\mathbf{x}) = \sum_{i=0}^{n-2}\sum_{j>1}^{n-1}\]

rac{1}{r_{ij}^{12}} - rac{1}{r_{ij}^{6}}

Where, in this exercise: .. math:

r_{ij} = \sqrt{(x_{3i}-x_{3j})^2 + (x_{3i+1}-x_{3j+1})^2) + (x_{3i+2}-x_{3j+2})^2}

Valid for any dimension, \(n = 3*k, k=2 , 3, 4, ..., 20\). \(k\) is the number of atoms in 3-D space constraints: unconstrained type: multi-modal with one global minimum; non-separable Value-to-reach: \(minima[k-2] + 0.0001\). Here, \(n\) represents the number of dimensions and \(x_i \in [-4, 4]\) for \(i = 1 ,..., n\). Global optimum: .. math:

ext{minima} = [-1.,-3.,-6.,-9.103852,-12.712062,-16.505384,\
         -19.821489, -24.113360, -28.422532,-32.765970,\
         -37.967600,-44.326801, -47.845157,-52.322627,\
         -56.815742,-61.317995, -66.530949, -72.659782,\
         -77.1777043]\

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f_{\\text{LennardJones}}(\\mathbf{x}) = \\sum_{i=0}^{n-2}\\sum_{j>1}^{n-1}\\frac{1}{r_{ij}^{12}} - \\frac{1}{r_{ij}^{6}}'

latex_formula_bounds = 'x_i \\in [-4, 4]'

latex_formula_dimension = 'd \\in [6:60]'

latex_formula_global_optimum = 'f = [-1.,-3.,-6.,-9.103852,-12.712062,-16.505384, -19.821489, -24.113360, -28.422532,-32.765970, -37.967600,-44.326801, -47.845157,-52.322627, -56.815742,-61.317995, -66.530949, -72.659782, 77.1777043]'

linear = False

modality = True

name = 'LennardJones Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

class opfunu.name_based.l_func.Leon(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

\[f_{ ext{Leon}}(\mathbf{x}) = \left(1 - x_{1}\]

ight)^{2}

• 100 left(x_{2} - x_{1}^{2}

ight)^{2}

with \(x_i \in [-1.2, 1.2]\) for \(i = 1, 2\). Global optimum: \(f(x) = 0\) for \(x = [1, 1]\)

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f_{\\text{Leon}}(\\mathbf{x}) = \\left(1 - x_{1}\\right)^{2} + 100 \\left(x_{2} - x_{1}^{2} \\right)^{2}'

latex_formula_bounds = 'x_i \\in [-1.2, 1.2]'

latex_formula_dimension = 'd = 2'

latex_formula_global_optimum = 'f(1, 1) = 0'

linear = False

modality = False

name = 'Leon Function'

parametric = False

randomized_term = False

scalable = False

separable = True

unimodal = True

class opfunu.name_based.l_func.Levy03(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Mishra, S. Global Optimization by Differential Evolution and Particle Swarm Methods: Evaluation

on Some Benchmark Functions. Munich Personal RePEc Archive, 2006, 1005

\[f_{ ext{Levy03}}(\mathbf{x}) = \sin^2(\pi y_1)+\sum_{i=1}^{n-1}(y_i-1)^2[1+10\sin^2(\pi y_{i+1})]+(y_n-1)^2\]

\[y_i=1+\]

rac{x_i-1}{4}

Here, \(n\) represents the number of dimensions and \(x_i \in [-10, 10]\) for \(i=1,...,n\). Global optimum: \(f(x_i) = 0\) for \(x_i = 1\) for \(i=1,...,n\)

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f_{\\text{Levy03}}(\\mathbf{x}) = \\sin^2(\\pi y_1)+\\sum_{i=1}^{n-1}(y_i-1)^2[1+10\\sin^2(\\pi y_{i+1})]+(y_n-1)^2'

latex_formula_bounds = 'x_i \\in [-10, 10]'

latex_formula_dimension = 'd \\in N^+'

latex_formula_global_optimum = 'f(1,... 1) = 0'

linear = False

modality = True

name = 'Levy 3 Function'

parametric = False

randomized_term = False

scalable = True

separable = False

unimodal = False

class opfunu.name_based.l_func.Levy05(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Mishra, S. Global Optimization by Differential Evolution and Particle Swarm Methods: Evaluation

on Some Benchmark Functions. Munich Personal RePEc Archive, 2006, 1005

\[f_{ ext{Levy05}}(\mathbf{x}) = \sum_{i=1}^{5} i \cos \left[(i-1)x_1 + i\]

ight] imes sum_{j=1}^{5} j cos left[(j+1)x_2 + j ight] + (x_1 + 1.42513)^2 + (x_2 + 0.80032)^2

Here, \(n\) represents the number of dimensions and \(x_i \in [-10, 10]\) for \(i=1,...,n\). Global optimum: \(f(x_i) = -176.1375779\) for \(\mathbf{x} = [-1.30685, -1.42485]\).

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f(\\mathbf{x}) = \\sum_{i=1}^{5} i \\cos \\left[(i-1)x_1 + i \\right] \\times \\sum_{j=1}^{5} j \\cos \\left[(j+1)x_2 + j \\right] + (x_1 + 1.42513)^2 + (x_2 + 0.80032)^2'

latex_formula_bounds = 'x_i \\in [-10, 10]'

latex_formula_dimension = 'd = 2'

latex_formula_global_optimum = 'f(-1.30685, -1.42485) = -176.1375779'

linear = False

modality = True

name = 'Levy 5 Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

class opfunu.name_based.l_func.Levy13(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Mishra, S. Global Optimization by Differential Evolution and Particle Swarm Methods: Evaluation

on Some Benchmark Functions. Munich Personal RePEc Archive, 2006, 1005

\[f_{ ext{Levy13}}(x) = \left(x_{1} -1\]

ight)^{2} left[sin^{2}left(3 pi x_{2} ight) + 1 ight] + left(x_{2}

• 1

ight)^{2} left[sin^{2}left(2 pi x_{2} ight)+ 1 ight] + sin^{2}left(3 pi x_{1} ight)

with \(x_i \in [-10, 10]\) for \(i = 1, 2\). Global optimum: \(f(x) = 0\) for \(x = [1, 1]\)

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f_{\\text{Levy13}}(x) = \\left(x_{1} -1\\right)^{2} \\left[\\sin^{2}\\left(3 \\pi x_{2}\\right) + 1\\right] + \\left(x_{2} - 1\\right)^{2} \\left[\\sin^{2}\\left(2 \\pi x_{2}\\right)+ 1\\right] + \\sin^{2}\\left(3 \\pi x_{1}\\right)'

latex_formula_bounds = 'x_i \\in [-10, 10]'

latex_formula_dimension = 'd = 2'

latex_formula_global_optimum = 'f(1., 1.) = 0'

linear = False

modality = True

name = 'Levy 5 Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

opfunu.name_based.m_func module

class opfunu.name_based.m_func.Matyas(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

\[f_{ ext{Matyas}}(x) = 0.26(x_1^2 + x_2^2) - 0.48 x_1 x_2\]

Here \(x_i \in [-10, 10]\) for \(i = 1, 2\). Global optimum: \(f(x) = 0.0\)

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f_{\\text{Matyas}}(x) = 0.26(x_1^2 + x_2^2) - 0.48 x_1 x_2'

latex_formula_bounds = 'x_i \\in [-10, 10]'

latex_formula_dimension = 'd = 2'

latex_formula_global_optimum = 'f(0, 0) = 0'

linear = False

modality = False

name = 'Matyas Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = True

class opfunu.name_based.m_func.McCormick(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

\[f(x) = - x_{1} + 2 x_{2} + \left(x_{1} - x_{2}\]

ight)^{2} + sinleft(x_{1} + x_{2} ight) + 1

Here \(x_1 \in [-1.5, 4], x_2 \in [-3, 4]\) . Global optimum: \(f(x) = -1.913222954981037\)

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f(x) = - x_{1} + 2 x_{2} + \\left(x_{1} - x_{2}\\right)^{2} + \\sin\\left(x_{1} + x_{2}\\right) + 1'

latex_formula_bounds = 'x_1 \\in [-1.5, 4], x_2 \\in [-3, 4]'

latex_formula_dimension = 'd = 2'

latex_formula_global_optimum = 'f(-0.5471975602214493, -1.547197559268372) = -1.913222954981037'

linear = False

modality = False

name = 'McCormick Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

class opfunu.name_based.m_func.Meyer(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

https://www.itl.nist.gov/div898/strd/nls/data/mgh10.shtml

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f(x)'

latex_formula_bounds = 'x_1 \\in [0, 1], x_2 \\in [100, 1000], x_3 \\in [100, 500]'

latex_formula_dimension = 'd = 3'

latex_formula_global_optimum = 'f(5.6096364710e-3, 6.1813463463e2, 3.4522363462e2) = 8.7945855171e1'

linear = False

modality = False

name = 'Meyer Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

class opfunu.name_based.m_func.Michalewicz(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Adorio, E. MVF - “Multivariate Test Functions Library in C for

Unconstrained Global Optimization”, 2005

\[f(x) = - \sum_{i=1}^{2} \sin\left(x_i\]

ight) sin^{2 m}left( rac{i x_i^{2}}{pi} ight)

Here \(x_i \in [0, \pi]\). Global optimum: \(f(x) = -1.8013\)

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f(x) = - x_{1} + 2 x_{2} + \\left(x_{1} - x_{2}\\right)^{2} + \\sin\\left(x_{1} + x_{2}\\right) + 1'

latex_formula_bounds = 'x_i \\in [0, \\pi]`'

latex_formula_dimension = 'd = 2'

latex_formula_global_optimum = 'f(0, 0) = -1.8013'

linear = False

modality = False

name = 'McCormick Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

class opfunu.name_based.m_func.MieleCantrell(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

\[f(x) = (e^{-x_1} - x_2)^4 + 100(x_2 - x_3)^6 + an^4(x_3 - x_4) + x_1^8\]

Here \(x_i \in [-1, 1] for i \in [1, 4]\). Global optimum: \(f(x) = 0\)

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f(x) = (e^{-x_1} - x_2)^4 + 100(x_2 - x_3)^6 + \\tan^4(x_3 - x_4) + x_1^8'

latex_formula_bounds = 'x_i \\in [-1, 1] for i \\in [1, 4]'

latex_formula_dimension = 'd = 4'

latex_formula_global_optimum = 'f(0, 1, 1, 1) = 0'

linear = False

modality = False

name = 'Miele Cantrell Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

class opfunu.name_based.m_func.Mishra01(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

\[f(x) = (1 + x_n)^{x_n} x_n = n - \sum_{i=1}^{n-1} x_i\]

Here \(x_i \in [0, 1] for i \in [1, n]\). Global optimum: \(f(x) = 2\)

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f(x) = (1 + x_n)^{x_n}; x_n = n - \\sum_{i=1}^{n-1} x_i'

latex_formula_bounds = 'x_i \\in [0, 1] for i \\in [1, n]'

latex_formula_dimension = 'd = n'

latex_formula_global_optimum = 'f(1) = 2'

linear = False

modality = False

name = 'Mishra 1 Function'

parametric = False

randomized_term = False

scalable = True

separable = False

unimodal = False

class opfunu.name_based.m_func.Mishra02(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

\[f(x) = (1 + x_n)^{x_n} x_n = n - \sum_{i=1}^{n-1}\]

rac{(x_i + x_{i+1})}{2}

Here \(x_i \in [0, 1] for i \in [1, n]\). Global optimum: \(f(x) = 2\)

continuous = True

convex = False

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f(x) = (1 + x_n)^{x_n}; x_n = n - \\sum_{i=1}^{n-1} \\frac{(x_i + x_{i+1})}{2}'

latex_formula_bounds = 'x_i \\in [0, 1] for i \\in [1, n]'

latex_formula_dimension = 'd = n'

latex_formula_global_optimum = 'f(1) = 2'

linear = False

modality = False

name = 'Mishra 2 Function'

parametric = False

randomized_term = False

scalable = True

separable = False

unimodal = False

class opfunu.name_based.m_func.Mishra03(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

\[f(x) = \sqrt{\lvert \cos{\sqrt{\lvert x_1^2 + x_2^2\]

vert}} vert} + 0.01(x_1 + x_2)

Here \(x_i \in [0, 1] for i \in [1, n]\). Global optimum: \(f(-9.99378322, -9.99918927) = -0.19990562\)

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f(x) = \\sqrt{\\lvert \\cos{\\sqrt{\\lvert x_1^2 + x_2^2 \\rvert}} \\rvert} + 0.01(x_1 + x_2)'

latex_formula_bounds = 'x_i \\in [-10, 10] for i \\in [1, 2]'

latex_formula_dimension = 'd = 2'

latex_formula_global_optimum = 'f(-9.99378322, -9.99918927) = -0.19990562'

linear = False

modality = False

name = 'Mishra 3 Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

class opfunu.name_based.m_func.Mishra04(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

\[f(x) = \sqrt{\lvert \sin{\sqrt{\lvert x_1^2 + x_2^2\]

vert}} vert} + 0.01(x_1 + x_2)

Here \(x_i \in [-10, 10] for i \in [1, n]\). Global optimum: \(f(-8.71499636, -9.0533148) = -0.17767\)

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f(x) = \\sqrt{\\lvert \\sin{\\sqrt{\\lvert x_1^2 + x_2^2 \\rvert}} \\rvert} + 0.01(x_1 + x_2)'

latex_formula_bounds = 'x_i \\in [-10, 10] for i \\in [1, 2]'

latex_formula_dimension = 'd = 2'

latex_formula_global_optimum = 'f(-8.71499636, -9.0533148) = -0.17767'

linear = False

modality = False

name = 'Mishra 4 Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

class opfunu.name_based.m_func.Mishra05(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

\[f(x) = \left [ \sin^2 ((\cos(x_1) + \cos(x_2))^2) + \cos^2 ((\sin(x_1) + \sin(x_2))^2) + x_1\]

ight ]^2 + 0.01(x_1 + x_2)

Here \(x_i \in [-10, 10] for i \in [1, 2]\). Global optimum: \(f(-1.98682, -10) = -1.019829519930646\)

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f(x) = \\left [ \\sin^2 ((\\cos(x_1) + \\cos(x_2))^2) + \\cos^2 ((\\sin(x_1) + \\sin(x_2))^2) + x_1 \\right ]^2 + 0.01(x_1 + x_2)'

latex_formula_bounds = 'x_i \\in [-10, 10] for i \\in [1, 2]'

latex_formula_dimension = 'd = 2'

latex_formula_global_optimum = 'f(-1.98682, -10) = -1.019829519930646'

linear = False

modality = False

name = 'Mishra 5 Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

class opfunu.name_based.m_func.Mishra06(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

\[f(x) = -\log{\left [ \sin^2 ((\cos(x_1) + \cos(x_2))^2) - \cos^2 ((\sin(x_1) + \sin(x_2))^2) + x_1\]

ight ]^2} + 0.01 left[(x_1 -1)^2 + (x_2 - 1)^2 ight]

Here \(x_i \in [-10, 10] for i \in [1, 2]\). Global optimum: \(f(2.88631, 1.82326) = -2.28395\)

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f(x) = -\\log{\\left [ \\sin^2 ((\\cos(x_1) + \\cos(x_2))^2) - \\cos^2 ((\\sin(x_1) + \\sin(x_2))^2) + x_1 \\right ]^2} + 0.01 \\left[(x_1 -1)^2 + (x_2 - 1)^2 \\right]'

latex_formula_bounds = 'x_i \\in [-10, 10] for i \\in [1, 2]'

latex_formula_dimension = 'd = 2'

latex_formula_global_optimum = 'f(2.88631, 1.82326) = -2.28395'

linear = False

modality = False

name = 'Mishra 6 Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

class opfunu.name_based.m_func.Mishra07(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

\[f(x) = \left [\prod_{i=1}^{n} x_i - n!\]

ight]^2

Here \(x_i \in [-10, 10] for i \in [1, n]\). Global optimum: :math:`f(sqrt{n}) = 0, `

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f(x) = \\left [\\prod_{i=1}^{n} x_i - n! \\right]^2'

latex_formula_bounds = 'x_i \\in [-10, 10] \\forall i \\in [1, n]'

latex_formula_dimension = 'd = n'

latex_formula_global_optimum = 'f(\\sqrt{n}) = 0'

linear = False

modality = False

name = 'Mishra 7 Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

class opfunu.name_based.m_func.Mishra08(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

\[f(x) = 0.001 \left[\lvert x_1^{10} - 20x_1^9 + 180x_1^8 - 960 x_1^7 + 3360x_1^6 - 8064x_1^5 + 13340x_1^4 - 15360x_1^3 + 11520x_1^2 - 5120x_1 + 2624\]

vert lvert x_2^4 + 12x_2^3 + 54x_2^2 + 108x_2 + 81 vert ight]^2

Here \(x_i \in [-10, 10] for i \in [1, 2]\). Global optimum: :math:`f(2, -3) = 0, `

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f(x) = 0.001 \\left[\\lvert x_1^{10} - 20x_1^9 + 180x_1^8 - 960 x_1^7 + 3360x_1^6 - 8064x_1^5 + 13340x_1^4 - 15360x_1^3 + 11520x_1^2 - 5120x_1 + 2624 \\rvert \\lvert x_2^4 + 12x_2^3 + 54x_2^2 + 108x_2 + 81 \\rvert \\right]^2'

latex_formula_bounds = 'x_i \\in [-10, 10] \\forall i \\in [1, 2]'

latex_formula_dimension = 'd = 2'

latex_formula_global_optimum = 'f(2, -3) = 0'

linear = False

modality = False

name = 'Mishra 8 Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

class opfunu.name_based.m_func.Mishra09(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

\[f(x) = \left[ ab^2c + abc^2 + b^2 + (x_1 + x_2 - x_3)^2\]

ight]^2

Where, in this exercise:

\[egin{cases} a = 2x_1^3 + 5x_1x_2 + 4x_3 - 2x_1^2x_3 - 18 \ b = x_1 + x_2^3 + x_1x_2^2 + x_1x_3^2 - 22 \ c = 8x_1^2 + 2x_2x_3 + 2x_2^2 + 3x_2^3 - 52 \end{cases}\]

Here \(x_i \in [-10, 10] for i \in [1, 2, 3]\). Global optimum: :math:`f(1, 2, 3) = 0, `

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = '\\left[ ab^2c + abc^2 + b^2 + (x_1 + x_2 - x_3)^2 \\right]^2'

latex_formula_bounds = 'x_i \\in [-10, 10] \\forall i \\in [1, 2, 3]'

latex_formula_dimension = 'd = 2'

latex_formula_global_optimum = 'f(1, 2, 3) = 0'

linear = False

modality = False

name = 'Mishra 9 Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

class opfunu.name_based.m_func.Mishra10(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

\[f(x) = \left[ \lfloor x_1 \perp x_2\]

floor - lfloor x_1 floor - lfloor x_2 floor ight]^2

Here \(x_i \in [-10, 10] for i \in [1, 2]\). Global optimum: :math:`f(2, 2) = 0, `

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = '\\left[ \\lfloor x_1 \\perp x_2 \\rfloor - \\lfloor x_1 \\rfloor - \\lfloor x_2 \\rfloor \\right]^2'

latex_formula_bounds = 'x_i \\in [-10, 10] \\forall i \\in [1, 2]'

latex_formula_dimension = 'd = 2'

latex_formula_global_optimum = 'f(2, 2) = 0'

linear = False

modality = False

name = 'Mishra 10 Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

class opfunu.name_based.m_func.Mishra11(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

\[f(x) = \left [\]

rac{1}{n} sum_{i=1}^{n} lvert x_i vert - left(prod_{i=1}^{n} lvert x_i vert ight )^{ rac{1}{n}} ight]^2

Here \(x_i \in [-10, 10] for i \in [1, 2]\). Global optimum: :math:`f(0) = 0, `

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = '\\left [ \\frac{1}{n} \\sum_{i=1}^{n} \\lvert x_i \\rvert - \\left(\\prod_{i=1}^{n} \\lvert x_i \\rvert \\right )^{\\frac{1}{n}} \\right]^2'

latex_formula_bounds = 'x_i \\in [-10, 10] \\forall i \\in [1, n]'

latex_formula_dimension = 'd = n'

latex_formula_global_optimum = 'f(0) = 0'

linear = False

modality = False

name = 'Mishra 11 Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

class opfunu.name_based.m_func.MultiModal(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Gavana, A. Global Optimization Benchmarks and AMPGO retrieved 2015

\[f(x) = \left( \sum_{i=1}^n \lvert x_i\]

vert ight) left( prod_{i=1}^n lvert x_i vert ight)

Here \(x_i \in [-10, 10] for i \in [1, n]\). Global optimum: :math:`f(0) = 0, `

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = '\\left( \\sum_{i=1}^n \\lvert x_i \\rvert \\right) \\left( \\prod_{i=1}^n \\lvert x_i \\rvert \\right)'

latex_formula_bounds = 'x_i \\in [-10, 10] \\forall i \\in [1, n]'

latex_formula_dimension = 'd = n'

latex_formula_global_optimum = 'f(0) = 0'

linear = False

modality = False

name = 'Mishra 11 Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

opfunu.name_based.n_func module

class opfunu.name_based.n_func.NeedleEye(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Gavana, A. Global Optimization Benchmarks and AMPGO retrieved 2015

\[f_{ ext{NeedleEye}}(x) = egin{cases} 1 & extrm{if }\hspace{5pt} \lvert x_i\]

vert < eye hspace{5pt}

orall i

sum_{i=1}^n (100 + lvert x_i

vert) & extrm{if } hspace{5pt}

lvert x_i

vert > eye

0 & extrm{otherwise}end{cases}

Here \(x_i \in [-10, 10]\) for \(i = 1, 2,...,n\). Global optimum: \(f(x) = 1.0\)

continuous = False

convex = True

differentiable = False

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f_{\\text{Matyas}}(x) = \\begin{cases} 1 & \\textrm{if }\\hspace{5pt} \\lvert x_i \\rvert  <  eye \\hspace{5pt} \\forall i \\\\ \\sum_{i=1}^n (100 + \\lvert x_i \\rvert) & \\textrm{if } \\hspace{5pt}\\lvert x_i \\rvert > eye \\\\ 0 & \\textrm{otherwise}\\\\\\end{cases}'

latex_formula_bounds = 'x_i \\in [-10, 10, ..., 10]'

latex_formula_dimension = 'd = n'

latex_formula_global_optimum = 'f(0, 0, ...,0) = 1.0'

linear = False

modality = False

name = 'NeedleEye Function'

parametric = False

randomized_term = False

scalable = True

separable = False

unimodal = False

class opfunu.name_based.n_func.NewFunction01(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Mishra, S. Global Optimization by Differential Evolution and

Particle Swarm Methods: Evaluation on Some Benchmark Functions. Munich Personal RePEc Archive, 2006, 1005

\[f_{ ext{NewFunction01}}(x) = \left | {\cos\left(\sqrt{\left|{x_{1}^{2} + x_{2}}\]

ight|} ight)} ight |^{0.5} + (x_{1} + x_{2})/100

Here \(x_i \in [-10, 10]\) for \(i = 1, 2\). Global optimum: \(f(x) = -0.18459899925\)

continuous = False

convex = True

differentiable = False

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f_{\text{NewFunction01}}(x) = \\left | {\\cos\\left(\\sqrt{\\left|{x_{1}^{2}+ x_{2}}\right|}\right)} \right |^{0.5} + (x_{1} + x_{2})/100'

latex_formula_bounds = 'x_i \\in [-10, 10]'

latex_formula_dimension = 'd = 2'

latex_formula_global_optimum = 'f([-8.46669057, -9.99982177]) = -0.18459899925'

linear = False

modality = False

name = 'NewFunction01 Function'

parametric = False

randomized_term = False

scalable = True

separable = False

unimodal = False

class opfunu.name_based.n_func.NewFunction02(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Mishra, S. Global Optimization by Differential Evolution and

Particle Swarm Methods: Evaluation on Some Benchmark Functions. Munich Personal RePEc Archive, 2006, 1005

\[f_{ ext{NewFunction02}}(x) = \left | {\sin\left(\sqrt{\lvert{x_{1}^{2} + x_{2}}\]

vert} ight)} ight |^{0.5} + (x_{1} + x_{2})/100

Here \(x_i \in [-10, 10]\) for \(i = 1, 2\). Global optimum: \(f(x) = -0.19933159253\)

continuous = False

convex = True

differentiable = False

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f_{\text{NewFunction02}}(x) = \\left | {\\sin\\left(\\sqrt{\\lvert{x_{1}^{2} + x_{2}}\rvert}\right)} \right |^{0.5} + (x_{1} + x_{2})/100'

latex_formula_bounds = 'x_i \\in [-10, 10]'

latex_formula_dimension = 'd = 2'

latex_formula_global_optimum = 'f([-9.94103375, -9.99771235]) = -0.19933159253'

linear = False

modality = False

name = 'NewFunction02 Function'

parametric = False

randomized_term = False

scalable = True

separable = False

unimodal = False

opfunu.name_based.o_func module

class opfunu.name_based.o_func.OddSquare(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Gavana, A. Global Optimization Benchmarks and AMPGO retrieved 2015

\[f_{ ext{OddSquare}}(x) = -e^{-\]

rac{d}{2pi}} cos(pi d) left( 1 + rac{0.02h}{d + 0.01} ight )

Where, in this exercise:

\[egin{cases} d = n \cdot \smash{\displaystyle\max_{1 \leq i \leq n}} \left[ (x_i - b_i)^2\]

ight ]

h = sum_{i=1}^{n} (x_i - b_i)^2 end{cases}

And :math:`b = [1, 1.3, 0.8, -0.4, -1.3, 1.6, -0.2, -0.6, 0.5, 1.4, 1, 1.3,

0.8, -0.4, -1.3, 1.6, -0.2, -0.6, 0.5, 1.4]`

Here \(x_i \in [-5 \pi, 5 \pi]\) for \(i = 1, ..., n\). n leq 20. Global optimum: \(f(x) = -1.00846728102\)

continuous = False

convex = True

differentiable = False

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f_{\\text{OddSquare}}(x) = -e^{-\\frac{d}{2\\pi}} \\cos(\\pi d) \\left( 1 + \\frac{0.02h}{d + 0.01} \\right )'

latex_formula_bounds = 'x_i \\in [-5 \\pi, 5 \\pi]'

latex_formula_dimension = 'd = 20'

latex_formula_global_optimum = 'f(b) = -1.00846728102'

linear = False

modality = False

name = 'Odd Square Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

opfunu.name_based.p_func module

class opfunu.name_based.p_func.Parsopoulos(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization Problems

Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

\[f_{ ext{Parsopoulos}}(x) = \cos(x_1)^2 + \sin(x_2)^2\]

with \(x_i \in [-5, 5]\) for \(i = 1, 2\).

Global optimum: This function has infinite number of global minima in R2, at points :math:`left(k

rac{pi}{2}, lambda pi ight)`, where \(k = \pm1, \pm3, ...\) and \(\lambda = 0, \pm1, \pm2, ...\)

In the given domain problem, function has 12 global minima all equal to zero.

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f_{\\text{Parsopoulos}}(x) = \\cos(x_1)^2 + \\sin(x_2)^2'

latex_formula_bounds = 'x_i \\in [-10, 10, ..., 10]'

latex_formula_dimension = 'd = n'

latex_formula_global_optimum = 'f(0, 0, ...,0) = 1.0'

linear = False

modality = False

name = 'Parsopoulos Function'

parametric = False

randomized_term = False

scalable = True

separable = True

unimodal = False

opfunu.name_based.q_func module

class opfunu.name_based.q_func.Qing(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization Problems

Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

\[f_{ ext{Qing}}(x) = \sum_{i=1}^{n} (x_i^2 - i)^2\]

Here, \(n\) represents the number of dimensions and \(x_i \in [-500, 500]\) for \(i = 1, ..., n\).

Global optimum: \(f(x) = 0\) for \(x_i = \pm \sqrt(i)\) for \(i = 1, ..., n\)

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f_{\\text{Qing}}(x) = \\sum_{i=1}^{n} (x_i^2 - i)^2'

latex_formula_bounds = 'x_i \\in [-10, 10, ..., 10]'

latex_formula_dimension = 'd = n'

latex_formula_global_optimum = 'f(0, 0, ...,0) = 1.0'

linear = False

modality = False

name = 'Qing Function'

parametric = False

randomized_term = False

scalable = True

separable = True

unimodal = False

class opfunu.name_based.q_func.Quadratic(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization Problems

Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

\[f_{ ext{Quadratic}}(x) = -3803.84 - 138.08x_1 - 232.92x_2 + 128.08x_1^2+ 203.64x_2^2 + 182.25x_1x_2\]

Here, \(n\) represents the number of dimensions and \(x_i \in [-10, 10]\) for \(i = 1, 2\).

Global optimum: \(f(x) = -3873.72418\) for \(x = [0.19388, 0.48513]\)

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f_{\\text{Quadratic}}(x) = -3803.84 - 138.08x_1 - 232.92x_2 + 128.08x_1^2+ 203.64x_2^2 + 182.25x_1x_2'

latex_formula_bounds = 'x_i \\in [-10, 10, ..., 10]'

latex_formula_dimension = 'd = n'

latex_formula_global_optimum = 'f(0, 0, ...,0) = 1.0'

linear = False

modality = False

name = 'Quadratic Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

class opfunu.name_based.q_func.Quartic(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization Problems

Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

\[f_{ ext{Quartic}}(x) =\]

Here, \(n\) represents the number of dimensions and \(x_i \in [-10, 10]\) for \(i = 1, 2\).

Global optimum: \(f(x) = -3873.72418\) for \(x = [0.19388, 0.48513]\)

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f_{\\text{Quartic}}(x) = '

latex_formula_bounds = 'x_i \\in [-10, 10, ..., 10]'

latex_formula_dimension = 'd = n'

latex_formula_global_optimum = 'f(0, 0, ...,0) = 1.0'

linear = False

modality = False

name = 'Quartic Function'

parametric = False

randomized_term = True

scalable = False

separable = False

unimodal = False

class opfunu.name_based.q_func.Quintic(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization Problems

Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

\[f_{ ext{Quintic}}(x) = \sum_{i=1}^{n} \left|{x_{i}^{5} - 3 x_{i}^{4}+ 4 x_{i}^{3} + 2 x_{i}^{2} - 10 x_{i} -4}\]

ight|

Here, \(n\) represents the number of dimensions and \(x_i \in [-10, 10]\) for \(i = 1, ..., n\).

Global optimum: \(f(x_i) = 0\) for \(x_i = -1\) for \(i = 1, ..., n\)

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f_{\\text{Quintic}}(x) = \\sum_{i=1}^{n} \\left|{x_{i}^{5} - 3 x_{i}^{4}+ 4 x_{i}^{3} + 2 x_{i}^{2} - 10 x_{i} -4}\\right|'

latex_formula_bounds = 'x_i \\in [-10, 10, ..., 10]'

latex_formula_dimension = 'd = n'

latex_formula_global_optimum = 'f(0, 0, ...,0) = 1.0'

linear = False

modality = True

name = 'Quartic Function'

parametric = False

randomized_term = False

scalable = True

separable = True

unimodal = False

opfunu.name_based.r_func module

class opfunu.name_based.r_func.Rana(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization Problems

Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

\[f_{ ext{Rana}}(x) = \sum_{i=1}^{n} \left[x_{i} \sin\left(\sqrt{\lvert{x_{1} - x_{i} + 1}\]

vert} ight)

cosleft(sqrt{lvert{x_{1} + x_{i} + 1}

vert} ight) + left(x_{1} + 1 ight) sinleft(sqrt{lvert{x_{1} + x_{i} +

1}

vert} ight) cosleft(sqrt{lvert{x_{1} - x_{i} +1} vert} ight) ight]

Here, \(n\) represents the number of dimensions and \(x_i \in [-500.0, 500.0]\) for \(i = 1, ..., n\).

Global optimum: \(f(x_i) = -928.5478\) for \(x = [-300.3376, 500]\).

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f_{\\text{Rana}}(x) = '

latex_formula_bounds = 'x_i \\in [-10, 10, ..., 10]'

latex_formula_dimension = 'd = n'

latex_formula_global_optimum = 'f(0, 0, ...,0) = 1.0'

linear = False

modality = True

name = 'Qing Function'

parametric = False

randomized_term = False

scalable = True

separable = False

unimodal = False

opfunu.name_based.s_func module

class opfunu.name_based.s_func.Salomon(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization Problems

Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

\[f_{ ext{Salomon}}(x) = 1 - \cos \left (2 \pi \sqrt{\sum_{i=1}^{n} x_i^2}\]

ight) + 0.1 sqrt{sum_{i=1}^n x_i^2}

Here, \(n\) represents the number of dimensions and \(x_i \in [-100, 100]\) for \(i = 1, ..., n\).

Global optimum: \(f(x) = 0\) for \(x_i = 0\) for \(i = 1, ..., n\)

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f_{\\text{Salomon}}(x) = 1 - \\cos \\left (2 \\pi \\sqrt{\\sum_{i=1}^{n} x_i^2} \\right) + 0.1 \\sqrt{\\sum_{i=1}^n x_i^2}'

latex_formula_bounds = 'x_i \\in [-10, 10, ..., 10]'

latex_formula_dimension = 'd = n'

latex_formula_global_optimum = 'f(0, 0, ...,0) = 1.0'

linear = False

modality = True

name = 'Qing Function'

parametric = False

randomized_term = False

scalable = True

separable = False

unimodal = False

opfunu.name_based.t_func module

class opfunu.name_based.t_func.TestTubeHolder(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization Problems

Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

\[f_{ ext{TestTubeHolder}}(x) = - 4 \left | {e^{\left|{\cos \left(\]

rac{1}{200} x_{1}^{2} +

rac{1}{200} x_{2}^{2} ight)} ight|}sinleft(x_{1} ight) cosleft(x_{2} ight)} ight|

with \(x_i \in [-10, 10]\) for \(i = 1, 2\).

Global optimum: \(f(x) = -10.872299901558\) for \(x= [-\pi/2, 0]\)

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f_{\\text{TestTubeHolder}}(x)='

latex_formula_bounds = 'x_i \\in [-10, 10, ..., 10]'

latex_formula_dimension = 'd = n'

latex_formula_global_optimum = 'f(0, 0, ...,0) = 1.0'

linear = False

modality = True

name = 'Qing Function'

parametric = False

randomized_term = False

scalable = False

separable = True

unimodal = False

opfunu.name_based.u_func module

class opfunu.name_based.u_func.Ursem01(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization Problems

Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

\[f_{ ext{Ursem01}}(x) = - \sin(2x_1 - 0.5 \pi) - 3 \cos(x_2) - 0.5 x_1\]

with \(x_1 \in [-2.5, 3]\) and \(x_2 \in [-2, 2]\).

Global optimum: \(f(x) = -4.81681406371\) for \(x = [1.69714, 0.0]\)

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f_{\\text{Ursem01}}(x) = - \\sin(2x_1 - 0.5 \\pi) - 3 \\cos(x_2) - 0.5 x_1'

latex_formula_bounds = 'x_i \\in [-10, 10, ..., 10]'

latex_formula_dimension = 'd = n'

latex_formula_global_optimum = 'f(0, 0, ...,0) = 1.0'

linear = False

modality = True

name = 'Qing Function'

parametric = False

randomized_term = False

scalable = False

separable = True

unimodal = False

opfunu.name_based.v_func module

class opfunu.name_based.v_func.VenterSobiezcczanskiSobieski(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization Problems

Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

\[f(x) = x_1^2 - 100 \cos^2(x_1) - 100 \cos(x_1^2/30)+ x_2^2 - 100 \cos^2(x_2)- 100 \cos(x_2^2/30)\]

with \(x_i \in [-50, 50]\) for \(i = 1, 2\).

Global optimum: \(f(x) = -400\) for \(x = [0, 0]\)

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f(x) = x_1^2 - 100 \\cos^2(x_1) - 100 \\cos(x_1^2/30)+ x_2^2 - 100 \\cos^2(x_2)- 100 \\cos(x_2^2/30)'

latex_formula_bounds = 'x_i \\in [-10, 10, ..., 10]'

latex_formula_dimension = 'd = n'

latex_formula_global_optimum = 'f(0, 0, ...,0) = 1.0'

linear = False

modality = True

name = 'VenterSobiezcczanskiSobieski Function'

parametric = False

randomized_term = False

scalable = False

separable = True

unimodal = False

opfunu.name_based.w_func module

class opfunu.name_based.w_func.Watson(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization Problems

Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

\[f(x) = \sum_{i=0}^{29} \left\{\sum_{j=0}^4 ((j + 1)a_i^j x_{j+1}) - \left[ \sum_{j=0}^5 a_i^j x_{j+1}\]

ight ]^2 - 1 ight}^2 + x_1^2

Where, in this exercise, \(a_i = i/29\). with \(x_i \in [-5, 5]\) for \(i = 1, ..., 6\).

Global optimum: \(f(x) = 0.002288\) for \(x = [-0.0158, 1.012, -0.2329, 1.260, -1.513, 0.9928]\)

continuous = True

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f(x) = \\sum_{i=0}^{29} \\left\\{\\sum_{j=0}^4 ((j + 1)a_i^j x_{j+1}) - \\left[ \\sum_{j=0}^5 a_i^j x_{j+1} \\right ]^2 - 1 \\right\\}^2 + x_1^2'

latex_formula_bounds = 'x_i \\in [-10, 10, ..., 10]'

latex_formula_dimension = 'd = n'

latex_formula_global_optimum = 'f(0, 0, ...,0) = 1.0'

linear = False

modality = True

name = 'Watson Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = True

opfunu.name_based.x_func module

class opfunu.name_based.x_func.XinSheYang01(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization Problems

Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

\[f(x) = \sum_{i=1}^{n} \epsilon_i \lvert x_i\]

vert^i

The variable \(\epsilon_i, (i = 1, ..., n)\) is a random variable uniformly distributed in \([0, 1]\).

Here, \(n\) represents the number of dimensions and \(x_i \in [-5, 5]\) for \(i = 1, ..., n\).

Global optimum: \(f(x) = 0\) for \(x_i = 0\) for \(i = 1, ..., n\)

continuous = True

convex = True

differentiable = False

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f(x) = \\sum_{i=1}^{n} \\epsilon_i \\lvert x_i \\rvert^i'

latex_formula_bounds = 'x_i \\in [-10, 10, ..., 10]'

latex_formula_dimension = 'd = n'

latex_formula_global_optimum = 'f(0, 0, ...,0) = 1.0'

linear = False

modality = True

name = 'Xin-She Yang 1 Function'

parametric = False

randomized_term = False

scalable = True

separable = True

unimodal = False

opfunu.name_based.y_func module

class opfunu.name_based.y_func.YaoLiu04(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Yao X., Liu Y. (1997) Fast evolution strategies. In: Angeline P.J., Reynolds R.G., McDonnell J.R., Eberhart R. (eds)

Evolutionary Programming VI. EP 1997. Lecture Notes in Computer Science, vol 1213. Springer, Berlin, Heidelberg

2

Mishra, S. Global Optimization by Differential Evolution and Particle Swarm Methods: Evaluation

on Some Benchmark Functions. Munich Personal RePEc Archive, 2006, 1005

\[f(x) = {max}_i \left\{ \left | x_i\]

ight | , 1 leq i leq n ight}

Here, \(n\) represents the number of dimensions and \(x_i \in [-10, 10]\) for \(i = 1, ..., n\).

Global optimum: \(f(x) = 0\) for \(x_i = 0\) for \(i = 1, ..., n\)

continuous = True

convex = True

differentiable = False

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f(x) = {max}_i \\left\\{ \\left | x_i \\right | , 1 \\leq i \\leq n \\right\\}'

latex_formula_bounds = 'x_i \\in [-10, 10, ..., 10]'

latex_formula_dimension = 'd = n'

latex_formula_global_optimum = 'f(0, 0, ...,0) = 1.0'

linear = False

modality = False

name = 'Yao-Liu 4 Function'

parametric = False

randomized_term = False

scalable = True

separable = True

unimodal = False

opfunu.name_based.z_func module

class opfunu.name_based.z_func.Zacharov(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization

Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

\[f_{ ext{Zacharov}}(x) = \sum_{i=1}^{n} x_i^2 + \left (\]

rac{1}{2}sum_{i=1}^{n} i x_i ight )^2

• left (

rac{1}{2} sum_{i=1}^{n} i x_i ight )^4

Here \(x_i \in [-5, 10]\) for \(i = 1, ..., n\). Global optimum: \(f(x) = 0.0\)

continuous = False

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = '\\sum_{i=1}^{n} x_i^2 + \\left ( \x0crac{1}{2}\\sum_{i=1}^{n} i x_i \right )^2+ \\left ( \x0crac{1}{2} \\sum_{i=1}^{n} i x_i \right )^4'

latex_formula_bounds = 'x_i \\in [-5, 10]'

latex_formula_dimension = 'd = n'

latex_formula_global_optimum = 'f(0, 0,...,0) = 0'

linear = False

modality = False

name = 'Zacharov Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

class opfunu.name_based.z_func.ZeroSum(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Gavana, A. Global Optimization Benchmarks and AMPGO retrieved 2015

\[f_{ ext{ZeroSum}}(x) = egin{cases} 0 & extrm{if} \sum_{i=1}^n x_i = 0 \ 1 + \left(10000 \left |\sum_{i=1}^n x_i\]

ight| ight)^{0.5} & extrm{otherwise} end{cases}

Here \(x_i \in [-10, 10]\) for \(i = 1, ..., n\). Global optimum: \(f(x) = 0.0\)sum_{i=1}^n x_i = 0`

continuous = False

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = '\x08egin{cases} 0 & \textrm{if} \\sum_{i=1}^n x_i = 0 \\ 1 + \\left(10000 \\left |\\sum_{i=1}^n x_i\right| \right)^{0.5} & \textrm{otherwise} \\end{cases}'

latex_formula_bounds = 'x_i \\in [-10, 10]'

latex_formula_dimension = 'd = n'

latex_formula_global_optimum = 'f(x_best) = 0'

linear = False

modality = False

name = 'ZeroSum Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

class opfunu.name_based.z_func.Zettl(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Gavana, A. Global Optimization Benchmarks and AMPGO retrieved 2015

\[f_{ ext{Zettl}}(x) =\]

rac{1}{4} x_{1} + left(x_{1}^{2} - 2 x_{1}

• x_{2}^{2}

ight)^{2}

Here \(x_i \in [-1, 5]\) for \(i = 1, 2\). Global optimum: \(f(x) = -0.0037912\)

continuous = False

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = '\\sum_{i=1}^{n} x_i^2 + \\left ( \x0crac{1}{2}\\sum_{i=1}^{n} i x_i \right )^2+ \\left ( \x0crac{1}{2} \\sum_{i=1}^{n} i x_i \right )^4'

latex_formula_bounds = 'x_i \\in [-0.029896, 0.0]'

latex_formula_dimension = 'd = 2'

latex_formula_global_optimum = 'f(x_best) = -0.0037912'

linear = False

modality = False

name = 'Zettl Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

class opfunu.name_based.z_func.Zimmerman(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Gavana, A. Global Optimization Benchmarks and AMPGO retrieved 2015

\[f_{ ext{Zimmerman}}(x) = \max \left[Zh1(x), Zp(Zh2(x)) extrm{sgn}(Zh2(x)), Zp(Zh3(x)) extrm{sgn}(Zh3(x)), Zp(-x_1) extrm{sgn}(x_1),Zp(-x_2) extrm{sgn}(x_2)\]

ight]

\[egin{cases} Zh1(x) = 9 - x_1 - x_2 \ Zh2(x) = (x_1 - 3)^2 + (x_2 - 2)^2 \ Zh3(x) = x_1x_2 - 14 \ Zp(t) = 100(1 + t) \end{cases}\]

Where \(x\) is a vector and \(t\) is a scalar. Here, \(x_i \in [0, 100]\) for \(i = 1, 2\). Global optimum: \(f(x) = 0\) for \(x = [7, 2]\)

continuous = False

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = '\\max \\left[Zh1(x), Zp(Zh2(x))\textrm{sgn}(Zh2(x)), Zp(Zh3(x))\textrm{sgn}(Zh3(x)), Zp(-x_1)\textrm{sgn}(x_1),Zp(-x_2)\textrm{sgn}(x_2) \right]'

latex_formula_bounds = 'x_i \\in [0, 100]'

latex_formula_dimension = 'd = 2'

latex_formula_global_optimum = 'f([7, 2]) = 0.'

linear = False

modality = False

name = 'Zimmerman Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = False

class opfunu.name_based.z_func.Zirilli(ndim=None, bounds=None)

Bases: opfunu.benchmark.Benchmark

1

Gavana, A. Global Optimization Benchmarks and AMPGO retrieved 2015

\[f_{ ext{Zirilli}}(x) = 0.25x_1^4 - 0.5x_1^2 + 0.1x_1 + 0.5x_2^2\]

\[egin{cases} Zh1(x) = 9 - x_1 - x_2 \ Zh2(x) = (x_1 - 3)^2 + (x_2 - 2)^2 \ Zh3(x) = x_1x_2 - 14 \ Zp(t) = 100(1 + t) \end{cases}\]

Where \(x\) is a vector and \(t\) is a scalar. Here, \(x_i \in [-10, 10]\) for \(i = 1, 2\). Global optimum: \(f(x) = -0.3523\) for \(x = [-1.0465, 0]\)

continuous = False

convex = True

differentiable = True

evaluate(x, *args)

Evaluation of the benchmark function.

Parameters

x (np.ndarray) – The candidate vector for evaluating the benchmark problem. Must have len(x) == self.ndim.

Returns

val – the evaluated benchmark function

Return type

float

latex_formula = 'f_{\text{Zirilli}}(x) = 0.25x_1^4 - 0.5x_1^2 + 0.1x_1 + 0.5x_2^2'

latex_formula_bounds = 'x_i \\in [-10, 10]'

latex_formula_dimension = 'd = 2'

latex_formula_global_optimum = 'f([-1.0465, 0]) = -0.3523'

linear = False

modality = False

name = 'Zirilli Function'

parametric = False

randomized_term = False

scalable = False

separable = False

unimodal = True

opfunu.utils package

opfunu.utils.operator module

opfunu.utils.operator.ackley_func(x)

opfunu.utils.operator.bent_cigar_func(x)

opfunu.utils.operator.calculate_weight(x, delta=1.0)

opfunu.utils.operator.chebyshev_func(x)

The following was converted from the cec2019 C code Storn’s Tchebychev - a 2nd ICEO function - generalized version

opfunu.utils.operator.different_powers_func(x)

opfunu.utils.operator.discus_func(x)

opfunu.utils.operator.doubledip(x, c, s)

opfunu.utils.operator.elliptic_func(x)

opfunu.utils.operator.expanded_griewank_rosenbrock_func(x)

This is based on the CEC version which unrolls the griewank and rosenbrock functions for better performance

opfunu.utils.operator.expanded_scaffer_f6_func(x)

opfunu.utils.operator.expanded_schaffer_f6_func(x)

This is a direct conversion of the CEC2021 C-Code for the Expanded Schaffer F6 Function

opfunu.utils.operator.f8f2_func(x)

opfunu.utils.operator.fractal_1d_func(x)

opfunu.utils.operator.generate_diagonal_matrix(size, alpha=10)

opfunu.utils.operator.grie_rosen_cec_func(x)

This is based on the CEC version which unrolls the griewank and rosenbrock functions for better performance

opfunu.utils.operator.griewank_func(x)

opfunu.utils.operator.gz_func(x)

opfunu.utils.operator.happy_cat_func(x, shift=0.0)

opfunu.utils.operator.hgbat_func(x, shift=0.0)

opfunu.utils.operator.inverse_hilbert_func(x)

This is a direct conversion of the cec2019 C code for python optimized to use numpy

opfunu.utils.operator.katsuura_func(x)

opfunu.utils.operator.lennard_jones_func(x)

This version is a direct python conversion from the C-Code of CEC2019 implementation. Find the atomic configuration with minimum energy (Lennard-Jones potential) Valid for any dimension, D = 3 * k, k = 2, 3, 4, …, 25. k is the number of atoms in 3-D space.

opfunu.utils.operator.levy_func(x, shift=0.0)

opfunu.utils.operator.lunacek_bi_rastrigin_func(x, miu0=2.5, d=1, shift=0.0)

opfunu.utils.operator.modified_schwefel_func(x)

This is a direct conversion of the CEC2021 C-Code for the Modified Schwefel F11 Function

opfunu.utils.operator.non_continuous_expanded_scaffer_func(x)

opfunu.utils.operator.non_continuous_rastrigin_func(x)

opfunu.utils.operator.rastrigin_func(x)

opfunu.utils.operator.rosenbrock_func(x, shift=0.0)

opfunu.utils.operator.rotated_expanded_scaffer_func(x)

opfunu.utils.operator.rotated_expanded_schaffer_func(x)

opfunu.utils.operator.rounder(x, condition)

opfunu.utils.operator.scaffer_func(x)

opfunu.utils.operator.schaffer_f7_func(x)

opfunu.utils.operator.schwefel_12_func(x)

opfunu.utils.operator.sphere_func(x)

opfunu.utils.operator.sphere_noise_func(x)

opfunu.utils.operator.tasy_func(x, beta=0.5)

opfunu.utils.operator.tosz_func(x)

opfunu.utils.operator.twist_func(x)

opfunu.utils.operator.weierstrass_func(x, a=0.5, b=3.0, k_max=20)

opfunu.utils.operator.weierstrass_norm_func(x, a=0.5, b=3.0, k_max=20)

This function matches CEC2005 description of F11 except for addition of the bias and follows the C implementation

opfunu.utils.operator.zakharov_func(x)

opfunu.utils.visualize module

opfunu.utils.visualize.plot_2d(func, n_space=1000, cmap=<matplotlib.colors.LinearSegmentedColormap object>, XYZ=None, ax=None, show=True)

opfunu.utils.visualize.plot_3d(func, n_space=1000, cmap=<matplotlib.colors.LinearSegmentedColormap object>, XYZ=None, ax=None, show=True)

opfunu.utils.visualize.plot_latex_formula(latex)

Cite Us

If you are using opfunu in your project, we would appreciate citations:

@software{thieu_nguyen_2020_3711682,
  author       = {Nguyen Van Thieu},
  title        = {Opfunu: An Open-source Python Library for Optimization Benchmark Functions},
  year         = 2020,
  publisher    = {Zenodo},
  doi          = {10.5281/zenodo.3620960},
  url          = {https://github.com/thieu1995/opfunu}
}

If you have an open-ended or a research question, you can contact me via: nguyenthieu2102@gmail.com

Important links

• Official source code repo: https://github.com/thieu1995/opfunu

• Official document: https://opfunu.readthedocs.io/

• Download releases: https://pypi.org/project/opfunu/

• Issue tracker: https://github.com/thieu1995/opfunu/issues

• Notable changes log: https://github.com/thieu1995/opfunu/blob/master/ChangeLog.md

• Examples with different version: https://github.com/thieu1995/opfunu/blob/master/examples.md

• Join our telegram community at [link](https://t.me/+fRVCJGuGJg1mNDg1)** to get more support and help.

• This project also related to our another projects which are optimization and machine learning. Check it here:

  • https://github.com/thieu1995/metaheuristics

  • https://github.com/thieu1995/mealpy

  • https://github.com/thieu1995/mafese

  • https://github.com/thieu1995/pfevaluator

  • https://github.com/thieu1995/MetaCluster

  • https://github.com/thieu1995/enoppy

  • https://github.com/thieu1995/permetrics

  • https://github.com/aiir-team

GNU General Public License

Version 3, 29 June 2007 Copyright © 2007 Free Software Foundation, Inc <http://fsf.org>

Everyone is permitted to copy and distribute verbatim copies of this license document, but changing it is not allowed.

Preamble

The GNU General Public License is a free, copyleft license for software and other kinds of works.

The licenses for most software and other practical works are designed to take away your freedom to share and change the works. By contrast, the GNU General Public License is intended to guarantee your freedom to share and change all versions of a program–to make sure it remains free software for all its users. We, the Free Software Foundation, use the GNU General Public License for most of our software; it applies also to any other work released this way by its authors. You can apply it to your programs, too.

When we speak of free software, we are referring to freedom, not price. Our General Public Licenses are designed to make sure that you have the freedom to distribute copies of free software (and charge for them if you wish), that you receive source code or can get it if you want it, that you can change the software or use pieces of it in new free programs, and that you know you can do these things.

To protect your rights, we need to prevent others from denying you these rights or asking you to surrender the rights. Therefore, you have certain responsibilities if you distribute copies of the software, or if you modify it: responsibilities to respect the freedom of others.

For example, if you distribute copies of such a program, whether gratis or for a fee, you must pass on to the recipients the same freedoms that you received. You must make sure that they, too, receive or can get the source code. And you must show them these terms so they know their rights.

Developers that use the GNU GPL protect your rights with two steps: (1) assert copyright on the software, and (2) offer you this License giving you legal permission to copy, distribute and/or modify it.

For the developers’ and authors’ protection, the GPL clearly explains that there is no warranty for this free software. For both users’ and authors’ sake, the GPL requires that modified versions be marked as changed, so that their problems will not be attributed erroneously to authors of previous versions.

Some devices are designed to deny users access to install or run modified versions of the software inside them, although the manufacturer can do so. This is fundamentally incompatible with the aim of protecting users’ freedom to change the software. The systematic pattern of such abuse occurs in the area of products for individuals to use, which is precisely where it is most unacceptable. Therefore, we have designed this version of the GPL to prohibit the practice for those products. If such problems arise substantially in other domains, we stand ready to extend this provision to those domains in future versions of the GPL, as needed to protect the freedom of users.

Finally, every program is threatened constantly by software patents. States should not allow patents to restrict development and use of software on general-purpose computers, but in those that do, we wish to avoid the special danger that patents applied to a free program could make it effectively proprietary. To prevent this, the GPL assures that patents cannot be used to render the program non-free.

The precise terms and conditions for copying, distribution and modification follow.

TERMS AND CONDITIONS

0. Definitions

“This License” refers to version 3 of the GNU General Public License.

“Copyright” also means copyright-like laws that apply to other kinds of works, such as semiconductor masks.

“The Program” refers to any copyrightable work licensed under this License. Each licensee is addressed as “you”. “Licensees” and “recipients” may be individuals or organizations.

To “modify” a work means to copy from or adapt all or part of the work in a fashion requiring copyright permission, other than the making of an exact copy. The resulting work is called a “modified version” of the earlier work or a work “based on” the earlier work.

A “covered work” means either the unmodified Program or a work based on the Program.

To “propagate” a work means to do anything with it that, without permission, would make you directly or secondarily liable for infringement under applicable copyright law, except executing it on a computer or modifying a private copy. Propagation includes copying, distribution (with or without modification), making available to the public, and in some countries other activities as well.

To “convey” a work means any kind of propagation that enables other parties to make or receive copies. Mere interaction with a user through a computer network, with no transfer of a copy, is not conveying.

An interactive user interface displays “Appropriate Legal Notices” to the extent that it includes a convenient and prominently visible feature that (1) displays an appropriate copyright notice, and (2) tells the user that there is no warranty for the work (except to the extent that warranties are provided), that licensees may convey the work under this License, and how to view a copy of this License. If the interface presents a list of user commands or options, such as a menu, a prominent item in the list meets this criterion.

1. Source Code

The “source code” for a work means the preferred form of the work for making modifications to it. “Object code” means any non-source form of a work.

A “Standard Interface” means an interface that either is an official standard defined by a recognized standards body, or, in the case of interfaces specified for a particular programming language, one that is widely used among developers working in that language.

The “System Libraries” of an executable work include anything, other than the work as a whole, that (a) is included in the normal form of packaging a Major Component, but which is not part of that Major Component, and (b) serves only to enable use of the work with that Major Component, or to implement a Standard Interface for which an implementation is available to the public in source code form. A “Major Component”, in this context, means a major essential component (kernel, window system, and so on) of the specific operating system (if any) on which the executable work runs, or a compiler used to produce the work, or an object code interpreter used to run it.

The “Corresponding Source” for a work in object code form means all the source code needed to generate, install, and (for an executable work) run the object code and to modify the work, including scripts to control those activities. However, it does not include the work’s System Libraries, or general-purpose tools or generally available free programs which are used unmodified in performing those activities but which are not part of the work. For example, Corresponding Source includes interface definition files associated with source files for the work, and the source code for shared libraries and dynamically linked subprograms that the work is specifically designed to require, such as by intimate data communication or control flow between those subprograms and other parts of the work.

The Corresponding Source need not include anything that users can regenerate automatically from other parts of the Corresponding Source.

The Corresponding Source for a work in source code form is that same work.

2. Basic Permissions

All rights granted under this License are granted for the term of copyright on the Program, and are irrevocable provided the stated conditions are met. This License explicitly affirms your unlimited permission to run the unmodified Program. The output from running a covered work is covered by this License only if the output, given its content, constitutes a covered work. This License acknowledges your rights of fair use or other equivalent, as provided by copyright law.

You may make, run and propagate covered works that you do not convey, without conditions so long as your license otherwise remains in force. You may convey covered works to others for the sole purpose of having them make modifications exclusively for you, or provide you with facilities for running those works, provided that you comply with the terms of this License in conveying all material for which you do not control copyright. Those thus making or running the covered works for you must do so exclusively on your behalf, under your direction and control, on terms that prohibit them from making any copies of your copyrighted material outside their relationship with you.

Conveying under any other circumstances is permitted solely under the conditions stated below. Sublicensing is not allowed; section 10 makes it unnecessary.

3. Protecting Users’ Legal Rights From Anti-Circumvention Law

No covered work shall be deemed part of an effective technological measure under any applicable law fulfilling obligations under article 11 of the WIPO copyright treaty adopted on 20 December 1996, or similar laws prohibiting or restricting circumvention of such measures.

When you convey a covered work, you waive any legal power to forbid circumvention of technological measures to the extent such circumvention is effected by exercising rights under this License with respect to the covered work, and you disclaim any intention to limit operation or modification of the work as a means of enforcing, against the work’s users, your or third parties’ legal rights to forbid circumvention of technological measures.

4. Conveying Verbatim Copies

You may convey verbatim copies of the Program’s source code as you receive it, in any medium, provided that you conspicuously and appropriately publish on each copy an appropriate copyright notice; keep intact all notices stating that this License and any non-permissive terms added in accord with section 7 apply to the code; keep intact all notices of the absence of any warranty; and give all recipients a copy of this License along with the Program.

You may charge any price or no price for each copy that you convey, and you may offer support or warranty protection for a fee.

5. Conveying Modified Source Versions

You may convey a work based on the Program, or the modifications to produce it from the Program, in the form of source code under the terms of section 4, provided that you also meet all of these conditions:

• a) The work must carry prominent notices stating that you modified it, and giving a relevant date.

• b) The work must carry prominent notices stating that it is released under this License and any conditions added under section 7. This requirement modifies the requirement in section 4 to “keep intact all notices”.

• c) You must license the entire work, as a whole, under this License to anyone who comes into possession of a copy. This License will therefore apply, along with any applicable section 7 additional terms, to the whole of the work, and all its parts, regardless of how they are packaged. This License gives no permission to license the work in any other way, but it does not invalidate such permission if you have separately received it.

• d) If the work has interactive user interfaces, each must display Appropriate Legal Notices; however, if the Program has interactive interfaces that do not display Appropriate Legal Notices, your work need not make them do so.

A compilation of a covered work with other separate and independent works, which are not by their nature extensions of the covered work, and which are not combined with it such as to form a larger program, in or on a volume of a storage or distribution medium, is called an “aggregate” if the compilation and its resulting copyright are not used to limit the access or legal rights of the compilation’s users beyond what the individual works permit. Inclusion of a covered work in an aggregate does not cause this License to apply to the other parts of the aggregate.

6. Conveying Non-Source Forms

You may convey a covered work in object code form under the terms of sections 4 and 5, provided that you also convey the machine-readable Corresponding Source under the terms of this License, in one of these ways:

• a) Convey the object code in, or embodied in, a physical product (including a physical distribution medium), accompanied by the Corresponding Source fixed on a durable physical medium customarily used for software interchange.

• b) Convey the object code in, or embodied in, a physical product (including a physical distribution medium), accompanied by a written offer, valid for at least three years and valid for as long as you offer spare parts or customer support for that product model, to give anyone who possesses the object code either (1) a copy of the Corresponding Source for all the software in the product that is covered by this License, on a durable physical medium customarily used for software interchange, for a price no more than your reasonable cost of physically performing this conveying of source, or (2) access to copy the Corresponding Source from a network server at no charge.

• c) Convey individual copies of the object code with a copy of the written offer to provide the Corresponding Source. This alternative is allowed only occasionally and noncommercially, and only if you received the object code with such an offer, in accord with subsection 6b.

• d) Convey the object code by offering access from a designated place (gratis or for a charge), and offer equivalent access to the Corresponding Source in the same way through the same place at no further charge. You need not require recipients to copy the Corresponding Source along with the object code. If the place to copy the object code is a network server, the Corresponding Source may be on a different server (operated by you or a third party) that supports equivalent copying facilities, provided you maintain clear directions next to the object code saying where to find the Corresponding Source. Regardless of what server hosts the Corresponding Source, you remain obligated to ensure that it is available for as long as needed to satisfy these requirements.

• e) Convey the object code using peer-to-peer transmission, provided you inform other peers where the object code and Corresponding Source of the work are being offered to the general public at no charge under subsection 6d.

A separable portion of the object code, whose source code is excluded from the Corresponding Source as a System Library, need not be included in conveying the object code work.

A “User Product” is either (1) a “consumer product”, which means any tangible personal property which is normally used for personal, family, or household purposes, or (2) anything designed or sold for incorporation into a dwelling. In determining whether a product is a consumer product, doubtful cases shall be resolved in favor of coverage. For a particular product received by a particular user, “normally used” refers to a typical or common use of that class of product, regardless of the status of the particular user or of the way in which the particular user actually uses, or expects or is expected to use, the product. A product is a consumer product regardless of whether the product has substantial commercial, industrial or non-consumer uses, unless such uses represent the only significant mode of use of the product.

“Installation Information” for a User Product means any methods, procedures, authorization keys, or other information required to install and execute modified versions of a covered work in that User Product from a modified version of its Corresponding Source. The information must suffice to ensure that the continued functioning of the modified object code is in no case prevented or interfered with solely because modification has been made.

If you convey an object code work under this section in, or with, or specifically for use in, a User Product, and the conveying occurs as part of a transaction in which the right of possession and use of the User Product is transferred to the recipient in perpetuity or for a fixed term (regardless of how the transaction is characterized), the Corresponding Source conveyed under this section must be accompanied by the Installation Information. But this requirement does not apply if neither you nor any third party retains the ability to install modified object code on the User Product (for example, the work has been installed in ROM).

The requirement to provide Installation Information does not include a requirement to continue to provide support service, warranty, or updates for a work that has been modified or installed by the recipient, or for the User Product in which it has been modified or installed. Access to a network may be denied when the modification itself materially and adversely affects the operation of the network or violates the rules and protocols for communication across the network.

Corresponding Source conveyed, and Installation Information provided, in accord with this section must be in a format that is publicly documented (and with an implementation available to the public in source code form), and must require no special password or key for unpacking, reading or copying.

7. Additional Terms

“Additional permissions” are terms that supplement the terms of this License by making exceptions from one or more of its conditions. Additional permissions that are applicable to the entire Program shall be treated as though they were included in this License, to the extent that they are valid under applicable law. If additional permissions apply only to part of the Program, that part may be used separately under those permissions, but the entire Program remains governed by this License without regard to the additional permissions.

When you convey a copy of a covered work, you may at your option remove any additional permissions from that copy, or from any part of it. (Additional permissions may be written to require their own removal in certain cases when you modify the work.) You may place additional permissions on material, added by you to a covered work, for which you have or can give appropriate copyright permission.

Notwithstanding any other provision of this License, for material you add to a covered work, you may (if authorized by the copyright holders of that material) supplement the terms of this License with terms:

• a) Disclaiming warranty or limiting liability differently from the terms of sections 15 and 16 of this License; or

• b) Requiring preservation of specified reasonable legal notices or author attributions in that material or in the Appropriate Legal Notices displayed by works containing it; or

• c) Prohibiting misrepresentation of the origin of that material, or requiring that modified versions of such material be marked in reasonable ways as different from the original version; or

• d) Limiting the use for publicity purposes of names of licensors or authors of the material; or

• e) Declining to grant rights under trademark law for use of some trade names, trademarks, or service marks; or

• f) Requiring indemnification of licensors and authors of that material by anyone who conveys the material (or modified versions of it) with contractual assumptions of liability to the recipient, for any liability that these contractual assumptions directly impose on those licensors and authors.

All other non-permissive additional terms are considered “further restrictions” within the meaning of section 10. If the Program as you received it, or any part of it, contains a notice stating that it is governed by this License along with a term that is a further restriction, you may remove that term. If a license document contains a further restriction but permits relicensing or conveying under this License, you may add to a covered work material governed by the terms of that license document, provided that the further restriction does not survive such relicensing or conveying.

If you add terms to a covered work in accord with this section, you must place, in the relevant source files, a statement of the additional terms that apply to those files, or a notice indicating where to find the applicable terms.

Additional terms, permissive or non-permissive, may be stated in the form of a separately written license, or stated as exceptions; the above requirements apply either way.

8. Termination

You may not propagate or modify a covered work except as expressly provided under this License. Any attempt otherwise to propagate or modify it is void, and will automatically terminate your rights under this License (including any patent licenses granted under the third paragraph of section 11).

However, if you cease all violation of this License, then your license from a particular copyright holder is reinstated (a) provisionally, unless and until the copyright holder explicitly and finally terminates your license, and (b) permanently, if the copyright holder fails to notify you of the violation by some reasonable means prior to 60 days after the cessation.

Moreover, your license from a particular copyright holder is reinstated permanently if the copyright holder notifies you of the violation by some reasonable means, this is the first time you have received notice of violation of this License (for any work) from that copyright holder, and you cure the violation prior to 30 days after your receipt of the notice.

Termination of your rights under this section does not terminate the licenses of parties who have received copies or rights from you under this License. If your rights have been terminated and not permanently reinstated, you do not qualify to receive new licenses for the same material under section 10.

9. Acceptance Not Required for Having Copies

You are not required to accept this License in order to receive or run a copy of the Program. Ancillary propagation of a covered work occurring solely as a consequence of using peer-to-peer transmission to receive a copy likewise does not require acceptance. However, nothing other than this License grants you permission to propagate or modify any covered work. These actions infringe copyright if you do not accept this License. Therefore, by modifying or propagating a covered work, you indicate your acceptance of this License to do so.

10. Automatic Licensing of Downstream Recipients

Each time you convey a covered work, the recipient automatically receives a license from the original licensors, to run, modify and propagate that work, subject to this License. You are not responsible for enforcing compliance by third parties with this License.

An “entity transaction” is a transaction transferring control of an organization, or substantially all assets of one, or subdividing an organization, or merging organizations. If propagation of a covered work results from an entity transaction, each party to that transaction who receives a copy of the work also receives whatever licenses to the work the party’s predecessor in interest had or could give under the previous paragraph, plus a right to possession of the Corresponding Source of the work from the predecessor in interest, if the predecessor has it or can get it with reasonable efforts.

You may not impose any further restrictions on the exercise of the rights granted or affirmed under this License. For example, you may not impose a license fee, royalty, or other charge for exercise of rights granted under this License, and you may not initiate litigation (including a cross-claim or counterclaim in a lawsuit) alleging that any patent claim is infringed by making, using, selling, offering for sale, or importing the Program or any portion of it.

11. Patents

A “contributor” is a copyright holder who authorizes use under this License of the Program or a work on which the Program is based. The work thus licensed is called the contributor’s “contributor version”.

A contributor’s “essential patent claims” are all patent claims owned or controlled by the contributor, whether already acquired or hereafter acquired, that would be infringed by some manner, permitted by this License, of making, using, or selling its contributor version, but do not include claims that would be infringed only as a consequence of further modification of the contributor version. For purposes of this definition, “control” includes the right to grant patent sublicenses in a manner consistent with the requirements of this License.

Each contributor grants you a non-exclusive, worldwide, royalty-free patent license under the contributor’s essential patent claims, to make, use, sell, offer for sale, import and otherwise run, modify and propagate the contents of its contributor version.

In the following three paragraphs, a “patent license” is any express agreement or commitment, however denominated, not to enforce a patent (such as an express permission to practice a patent or covenant not to sue for patent infringement). To “grant” such a patent license to a party means to make such an agreement or commitment not to enforce a patent against the party.

If you convey a covered work, knowingly relying on a patent license, and the Corresponding Source of the work is not available for anyone to copy, free of charge and under the terms of this License, through a publicly available network server or other readily accessible means, then you must either (1) cause the Corresponding Source to be so available, or (2) arrange to deprive yourself of the benefit of the patent license for this particular work, or (3) arrange, in a manner consistent with the requirements of this License, to extend the patent license to downstream recipients. “Knowingly relying” means you have actual knowledge that, but for the patent license, your conveying the covered work in a country, or your recipient’s use of the covered work in a country, would infringe one or more identifiable patents in that country that you have reason to believe are valid.

If, pursuant to or in connection with a single transaction or arrangement, you convey, or propagate by procuring conveyance of, a covered work, and grant a patent license to some of the parties receiving the covered work authorizing them to use, propagate, modify or convey a specific copy of the covered work, then the patent license you grant is automatically extended to all recipients of the covered work and works based on it.

A patent license is “discriminatory” if it does not include within the scope of its coverage, prohibits the exercise of, or is conditioned on the non-exercise of one or more of the rights that are specifically granted under this License. You may not convey a covered work if you are a party to an arrangement with a third party that is in the business of distributing software, under which you make payment to the third party based on the extent of your activity of conveying the work, and under which the third party grants, to any of the parties who would receive the covered work from you, a discriminatory patent license (a) in connection with copies of the covered work conveyed by you (or copies made from those copies), or (b) primarily for and in connection with specific products or compilations that contain the covered work, unless you entered into that arrangement, or that patent license was granted, prior to 28 March 2007.

Nothing in this License shall be construed as excluding or limiting any implied license or other defenses to infringement that may otherwise be available to you under applicable patent law.

12. No Surrender of Others’ Freedom

If conditions are imposed on you (whether by court order, agreement or otherwise) that contradict the conditions of this License, they do not excuse you from the conditions of this License. If you cannot convey a covered work so as to satisfy simultaneously your obligations under this License and any other pertinent obligations, then as a consequence you may not convey it at all. For example, if you agree to terms that obligate you to collect a royalty for further conveying from those to whom you convey the Program, the only way you could satisfy both those terms and this License would be to refrain entirely from conveying the Program.

13. Use with the GNU Affero General Public License

Notwithstanding any other provision of this License, you have permission to link or combine any covered work with a work licensed under version 3 of the GNU Affero General Public License into a single combined work, and to convey the resulting work. The terms of this License will continue to apply to the part which is the covered work, but the special requirements of the GNU Affero General Public License, section 13, concerning interaction through a network will apply to the combination as such.

14. Revised Versions of this License

The Free Software Foundation may publish revised and/or new versions of the GNU General Public License from time to time. Such new versions will be similar in spirit to the present version, but may differ in detail to address new problems or concerns.

Each version is given a distinguishing version number. If the Program specifies that a certain numbered version of the GNU General Public License “or any later version” applies to it, you have the option of following the terms and conditions either of that numbered version or of any later version published by the Free Software Foundation. If the Program does not specify a version number of the GNU General Public License, you may choose any version ever published by the Free Software Foundation.

If the Program specifies that a proxy can decide which future versions of the GNU General Public License can be used, that proxy’s public statement of acceptance of a version permanently authorizes you to choose that version for the Program.

Later license versions may give you additional or different permissions. However, no additional obligations are imposed on any author or copyright holder as a result of your choosing to follow a later version.

15. Disclaimer of Warranty

THERE IS NO WARRANTY FOR THE PROGRAM, TO THE EXTENT PERMITTED BY APPLICABLE LAW. EXCEPT WHEN OTHERWISE STATED IN WRITING THE COPYRIGHT HOLDERS AND/OR OTHER PARTIES PROVIDE THE PROGRAM “AS IS” WITHOUT WARRANTY OF ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE. THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE PROGRAM IS WITH YOU. SHOULD THE PROGRAM PROVE DEFECTIVE, YOU ASSUME THE COST OF ALL NECESSARY SERVICING, REPAIR OR CORRECTION.

16. Limitation of Liability

IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRITING WILL ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MODIFIES AND/OR CONVEYS THE PROGRAM AS PERMITTED ABOVE, BE LIABLE TO YOU FOR DAMAGES, INCLUDING ANY GENERAL, SPECIAL, INCIDENTAL OR CONSEQUENTIAL DAMAGES ARISING OUT OF THE USE OR INABILITY TO USE THE PROGRAM (INCLUDING BUT NOT LIMITED TO LOSS OF DATA OR DATA BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY YOU OR THIRD PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH ANY OTHER PROGRAMS), EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE POSSIBILITY OF SUCH DAMAGES.

17. Interpretation of Sections 15 and 16

If the disclaimer of warranty and limitation of liability provided above cannot be given local legal effect according to their terms, reviewing courts shall apply local law that most closely approximates an absolute waiver of all civil liability in connection with the Program, unless a warranty or assumption of liability accompanies a copy of the Program in return for a fee.

END OF TERMS AND CONDITIONS

How to Apply These Terms to Your New Programs

If you develop a new program, and you want it to be of the greatest possible use to the public, the best way to achieve this is to make it free software which everyone can redistribute and change under these terms.

To do so, attach the following notices to the program. It is safest to attach them to the start of each source file to most effectively state the exclusion of warranty; and each file should have at least the “copyright” line and a pointer to where the full notice is found.

<one line to give the program’s name and a brief idea of what it does.> Copyright (C) <year> <name of author>

This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version.

This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with this program. If not, see <http://www.gnu.org/licenses/>.

Also add information on how to contact you by electronic and paper mail.

If the program does terminal interaction, make it output a short notice like this when it starts in an interactive mode:

<program> Copyright (C) <year> <name of author> This program comes with ABSOLUTELY NO WARRANTY; for details type ‘show w’. This is free software, and you are welcome to redistribute it under certain conditions; type ‘show c’ for details.

The hypothetical commands show w and show c should show the appropriate parts of the General Public License. Of course, your program’s commands might be different; for a GUI interface, you would use an “about box”.

You should also get your employer (if you work as a programmer) or school, if any, to sign a “copyright disclaimer” for the program, if necessary. For more information on this, and how to apply and follow the GNU GPL, see <http://www.gnu.org/licenses/>.

The GNU General Public License does not permit incorporating your program into proprietary programs. If your program is a subroutine library, you may consider it more useful to permit linking proprietary applications with the library. If this is what you want to do, use the GNU Lesser General Public License instead of this License. But first, please read <http://www.gnu.org/philosophy/why-not-lgpl.html>.

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