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