#!/usr/bin/env python
# Created by "Thieu" at 17:31, 30/07/2022 ----------%
# Email: nguyenthieu2102@gmail.com %
# Github: https://github.com/thieu1995 %
# --------------------------------------------------%
import numpy as np
from opfunu.benchmark import Benchmark
[docs]class Langermann(Benchmark):
"""
.. [1] Gavana, A. Global Optimization Benchmarks and AMPGO retrieved 2015
.. math::
f_{\text{Langermann}}(x) = - \sum_{i=1}^{5}
\frac{c_i \cos\left\{\pi \left[\left(x_{1}- a_i\right)^{2}
+ \left(x_{2} - b_i \right)^{2}\right]\right\}}{e^{\frac{\left( x_{1}
- a_i\right)^{2} + \left( x_{2} - b_i\right)^{2}}{\pi}}}
Where:
.. math::
\begin{matrix}
a = [3, 5, 2, 1, 7]\\
b = [5, 2, 1, 4, 9]\\
c = [1, 2, 5, 2, 3] \\
\end{matrix}
Here :math:`x_i \in [0, 10]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x) = -5.1621259`for :math:`x = [2.00299219, 1.006096]`
"""
name = "Langermann Function"
latex_formula = r'f_{\text{Langermann}}(x) = - \sum_{i=1}^{5} \frac{c_i \cos\left\{\pi \left[\left(x_{1}- a_i\right)^{2} + \left(x_{2} - b_i \right)^{2}\right]\right\}}{e^{\frac{\left( x_{1} - a_i\right)^{2} + \left( x_{2} - b_i\right)^{2}}{\pi}}}'
latex_formula_dimension = r'd = 2'
latex_formula_bounds = r'x_i \in [0, 10]'
latex_formula_global_optimum = r'f(2.00299219, 1.006096) = -5.1621259'
continuous = True
linear = False
convex = False
unimodal = False
separable = False
differentiable = True
scalable = False
randomized_term = False
parametric = False
modality = True # Number of ambiguous peaks, unknown # peaks
def __init__(self, ndim=None, bounds=None):
super().__init__()
self.dim_changeable = False
self.dim_default = 2
self.check_ndim_and_bounds(ndim, bounds, np.array([[0., 10.] for _ in range(self.dim_default)]))
self.f_global = -5.1621259
self.x_global = np.array([2.00299219, 1.006096])
self.a = np.array([3, 5, 2, 1, 7])
self.b = np.array([5, 2, 1, 4, 9])
self.c = np.array([1, 2, 5, 2, 3])
[docs] def evaluate(self, x, *args):
self.check_solution(x)
self.n_fe += 1
return (-np.sum(self.c * np.exp(-(1 / np.pi) * ((x[0] - self.a) ** 2 +
(x[1] - self.b) ** 2)) * np.cos(np.pi * ((x[0] - self.a) ** 2 + (x[1] - self.b) ** 2))))
[docs]class LennardJones(Benchmark):
"""
.. [1] http://www-wales.ch.cam.ac.uk/~jon/structures/LJ/tables.150.html
.. math::
f_{\text{LennardJones}}(\mathbf{x}) = \sum_{i=0}^{n-2}\sum_{j>1}^{n-1}
\frac{1}{r_{ij}^{12}} - \frac{1}{r_{ij}^{6}}
Where, in this exercise:
.. math::
r_{ij} = \sqrt{(x_{3i}-x_{3j})^2 + (x_{3i+1}-x_{3j+1})^2) + (x_{3i+2}-x_{3j+2})^2}
Valid for any dimension, :math:`n = 3*k, k=2 , 3, 4, ..., 20`. :math:`k` is the number of atoms in 3-D space
constraints: unconstrained type: multi-modal with one global minimum; non-separable Value-to-reach: :math:`minima[k-2] + 0.0001`.
Here, :math:`n` represents the number of dimensions and :math:`x_i \in [-4, 4]` for :math:`i = 1 ,..., n`.
*Global optimum*:
.. math::
\text{minima} = [-1.,-3.,-6.,-9.103852,-12.712062,-16.505384,\\
-19.821489, -24.113360, -28.422532,-32.765970,\\
-37.967600,-44.326801, -47.845157,-52.322627,\\
-56.815742,-61.317995, -66.530949, -72.659782,\\
-77.1777043]\\
"""
name = "LennardJones Function"
latex_formula = r'f_{\text{LennardJones}}(\mathbf{x}) = \sum_{i=0}^{n-2}\sum_{j>1}^{n-1}\frac{1}{r_{ij}^{12}} - \frac{1}{r_{ij}^{6}}'
latex_formula_dimension = r'd \in [6:60]'
latex_formula_bounds = r'x_i \in [-4, 4]'
latex_formula_global_optimum = r'f = [-1.,-3.,-6.,-9.103852,-12.712062,-16.505384, -19.821489, -24.113360, -28.422532,-32.765970, -37.967600,' \
r'-44.326801, -47.845157,-52.322627, -56.815742,-61.317995, -66.530949, -72.659782, 77.1777043]'
continuous = True
linear = False
convex = False
unimodal = False
separable = False
differentiable = True
scalable = False
randomized_term = False
parametric = False
modality = True # Number of ambiguous peaks, unknown # peaks
def __init__(self, ndim=None, bounds=None):
super().__init__()
self.dim_supported = list(range(6, 61))
self.dim_changeable = True
self.dim_default = 6
self.check_ndim_and_bounds(ndim, bounds, np.array([[-4., 4.] for _ in range(self.dim_default)]))
self.minima = [-1.0, -3.0, -6.0, -9.103852, -12.712062,
-16.505384, -19.821489, -24.113360, -28.422532,
-32.765970, -37.967600, -44.326801, -47.845157,
-52.322627, -56.815742, -61.317995, -66.530949,
-72.659782, -77.1777043]
self.f_global = self.minima[int(self.ndim/3) - 2]
self.x_global = np.zeros(self.ndim)
[docs] def evaluate(self, x, *args):
if self.ndim not in self.dim_supported:
raise ValueError(f"{self.__class__.__name__} problem is only supported ndim in {self.dim_supported}!")
self.check_solution(x)
self.n_fe += 1
k = int(self.ndim / 3)
s = 0.0
for i in range(k - 1):
for j in range(i + 1, k):
a = 3 * i
b = 3 * j
xd = x[a] - x[b]
yd = x[a + 1] - x[b + 1]
zd = x[a + 2] - x[b + 2]
ed = xd * xd + yd * yd + zd * zd
ud = ed * ed * ed
if ed > 0.0:
s += (1.0 / ud - 2.0) / ud
return s
[docs]class Leon(Benchmark):
"""
.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization
Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.
.. math::
f_{\text{Leon}}(\mathbf{x}) = \left(1 - x_{1}\right)^{2}
+ 100 \left(x_{2} - x_{1}^{2} \right)^{2}
with :math:`x_i \in [-1.2, 1.2]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x) = 0` for :math:`x = [1, 1]`
"""
name = "Leon Function"
latex_formula = r'f_{\text{Leon}}(\mathbf{x}) = \left(1 - x_{1}\right)^{2} + 100 \left(x_{2} - x_{1}^{2} \right)^{2}'
latex_formula_dimension = r'd = 2'
latex_formula_bounds = r'x_i \in [-1.2, 1.2]'
latex_formula_global_optimum = r'f(1, 1) = 0'
continuous = True
linear = False
convex = True
unimodal = True
separable = True
differentiable = True
scalable = False
randomized_term = False
parametric = False
modality = False # Number of ambiguous peaks, unknown # peaks
def __init__(self, ndim=None, bounds=None):
super().__init__()
self.dim_changeable = False
self.dim_default = 2
self.check_ndim_and_bounds(ndim, bounds, np.array([[-1.2, 1.2] for _ in range(self.dim_default)]))
self.f_global = 0.0
self.x_global = np.ones(self.ndim)
[docs] def evaluate(self, x, *args):
self.check_solution(x)
self.n_fe += 1
return 100. * (x[1] - x[0] ** 2.0) ** 2.0 + (1 - x[0]) ** 2.0
[docs]class Levy03(Benchmark):
"""
.. [1] Mishra, S. Global Optimization by Differential Evolution and Particle Swarm Methods: Evaluation
on Some Benchmark Functions. Munich Personal RePEc Archive, 2006, 1005
.. math::
f_{\text{Levy03}}(\mathbf{x}) = \sin^2(\pi y_1)+\sum_{i=1}^{n-1}(y_i-1)^2[1+10\sin^2(\pi y_{i+1})]+(y_n-1)^2
.. math::
y_i=1+\frac{x_i-1}{4}
Here, :math:`n` represents the number of dimensions and :math:`x_i \in [-10, 10]` for :math:`i=1,...,n`.
*Global optimum*: :math:`f(x_i) = 0` for :math:`x_i = 1` for :math:`i=1,...,n`
"""
name = "Levy 3 Function"
latex_formula = r'f_{\text{Levy03}}(\mathbf{x}) = \sin^2(\pi y_1)+\sum_{i=1}^{n-1}(y_i-1)^2[1+10\sin^2(\pi y_{i+1})]+(y_n-1)^2'
latex_formula_dimension = r'd \in N^+'
latex_formula_bounds = r'x_i \in [-10, 10]'
latex_formula_global_optimum = r'f(1,... 1) = 0'
continuous = True
linear = False
convex = True
unimodal = False
separable = False
differentiable = True
scalable = True
randomized_term = False
parametric = False
modality = True # Number of ambiguous peaks, unknown # peaks
def __init__(self, ndim=None, bounds=None):
super().__init__()
self.dim_changeable = True
self.dim_default = 2
self.check_ndim_and_bounds(ndim, bounds, np.array([[-10., 10.] for _ in range(self.dim_default)]))
self.f_global = 0.0
self.x_global = np.ones(self.ndim)
[docs] def evaluate(self, x, *args):
self.check_solution(x)
self.n_fe += 1
y = 1 + (x - 1) / 4
v = np.sum((y[:-1] - 1) ** 2 * (1 + 10 * np.sin(np.pi * y[1:]) ** 2))
z = (y[-1] - 1) ** 2
return np.sin(np.pi * y[0]) ** 2 + v + z
[docs]class Levy05(Benchmark):
"""
.. [1] Mishra, S. Global Optimization by Differential Evolution and Particle Swarm Methods: Evaluation
on Some Benchmark Functions. Munich Personal RePEc Archive, 2006, 1005
.. math::
f_{\text{Levy05}}(\mathbf{x}) = \sum_{i=1}^{5} i \cos \left[(i-1)x_1 + i \right] \times \sum_{j=1}^{5} j \cos \left[(j+1)x_2 + j \right] + (x_1 + 1.42513)^2 + (x_2 + 0.80032)^2
Here, :math:`n` represents the number of dimensions and :math:`x_i \in [-10, 10]` for :math:`i=1,...,n`.
*Global optimum*: :math:`f(x_i) = -176.1375779` for :math:`\mathbf{x} = [-1.30685, -1.42485]`.
"""
name = "Levy 5 Function"
latex_formula = r'f(\mathbf{x}) = \sum_{i=1}^{5} i \cos \left[(i-1)x_1 + i \right] \times \sum_{j=1}^{5} j \cos \left[(j+1)x_2 + j \right] + (x_1 + 1.42513)^2 + (x_2 + 0.80032)^2'
latex_formula_dimension = r'd = 2'
latex_formula_bounds = r'x_i \in [-10, 10]'
latex_formula_global_optimum = r'f(-1.30685, -1.42485) = -176.1375779'
continuous = True
linear = False
convex = False
unimodal = False
separable = False
differentiable = True
scalable = False
randomized_term = False
parametric = False
modality = True # Number of ambiguous peaks, unknown # peaks
def __init__(self, ndim=None, bounds=None):
super().__init__()
self.dim_changeable = False
self.dim_default = 2
self.check_ndim_and_bounds(ndim, bounds, np.array([[-10., 10.] for _ in range(self.dim_default)]))
self.f_global = -176.1375779
self.x_global = np.array([-1.30685, -1.42485])
[docs] def evaluate(self, x, *args):
self.check_solution(x)
self.n_fe += 1
idx = np.arange(1, 6)
a = idx * np.cos((idx - 1) * x[0] + idx)
b = idx * np.cos((idx + 1) * x[1] + idx)
return np.sum(a) * np.sum(b) + (x[0] + 1.42513) ** 2 + (x[1] + 0.80032) ** 2
[docs]class Levy13(Benchmark):
"""
.. [1] Mishra, S. Global Optimization by Differential Evolution and Particle Swarm Methods: Evaluation
on Some Benchmark Functions. Munich Personal RePEc Archive, 2006, 1005
.. math::
f_{\text{Levy13}}(x) = \left(x_{1} -1\right)^{2} \left[\sin^{2}\left(3 \pi x_{2}\right) + 1\right] + \left(x_{2}
- 1\right)^{2} \left[\sin^{2}\left(2 \pi x_{2}\right)+ 1\right] + \sin^{2}\left(3 \pi x_{1}\right)
with :math:`x_i \in [-10, 10]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x) = 0` for :math:`x = [1, 1]`
"""
name = "Levy 5 Function"
latex_formula = r'f_{\text{Levy13}}(x) = \left(x_{1} -1\right)^{2} \left[\sin^{2}\left(3 \pi x_{2}\right) + 1\right] + \left(x_{2} - 1\right)^{2} \left[\sin^{2}\left(2 \pi x_{2}\right)+ 1\right] + \sin^{2}\left(3 \pi x_{1}\right)'
latex_formula_dimension = r'd = 2'
latex_formula_bounds = r'x_i \in [-10, 10]'
latex_formula_global_optimum = r'f(1., 1.) = 0'
continuous = True
linear = False
convex = True
unimodal = False
separable = False
differentiable = True
scalable = False
randomized_term = False
parametric = False
modality = True # Number of ambiguous peaks, unknown # peaks
def __init__(self, ndim=None, bounds=None):
super().__init__()
self.dim_changeable = False
self.dim_default = 2
self.check_ndim_and_bounds(ndim, bounds, np.array([[-10., 10.] for _ in range(self.dim_default)]))
self.f_global = 0.
self.x_global = np.array([1., 1.])
[docs] def evaluate(self, x, *args):
self.check_solution(x)
self.n_fe += 1
u = np.sin(3 * np.pi * x[0]) ** 2
v = (x[0] - 1) ** 2 * (1 + (np.sin(3 * np.pi * x[1])) ** 2)
w = (x[1] - 1) ** 2 * (1 + (np.sin(2 * np.pi * x[1])) ** 2)
return u + v + w