#!/usr/bin/env python
# Created by "Thieu" at 17:55, 22/07/2022 ----------%
# Email: nguyenthieu2102@gmail.com %
# Github: https://github.com/thieu1995 %
# --------------------------------------------------%
import numpy as np
from opfunu.benchmark import Benchmark
[docs]class Hansen(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.
"""
name = "Hansen Function"
latex_formula = r'f(x) = \left[ \sum_{i=0}^4(i+1)\cos(ix_1+i+1)\right ]\left[\sum_{j=0}^4(j+1)\cos[(j+2)x_2+j+1])\right ]'
latex_formula_dimension = r'd = 2'
latex_formula_bounds = r'x_i \in [-10, 10], \forall i \in \llbracket 1, d\rrbracket'
latex_formula_global_optimum = r'f(-7.58989583, -7.70831466) = -176.54179'
continuous = True
linear = False
convex = True
unimodal = False
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([[-10., 10.] for _ in range(self.dim_default)]))
self.f_global = -176.54179
self.x_global = np.array([-7.58989583, -7.70831466])
[docs] def evaluate(self, x, *args):
self.check_solution(x)
self.n_fe += 1
i = np.arange(5.)
a = (i + 1) * np.cos(i * x[0] + i + 1)
b = (i + 1) * np.cos((i + 2) * x[1] + i + 1)
return np.sum(a) * np.sum(b)
[docs]class Hartmann3(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.
"""
name = "Hartman 3 Function"
latex_formula = r'f(x) = -\sum\limits_{i=1}^{4} c_i e^{-\sum\limits_{j=1}^{n}a_{ij}(x_j - p_{ij})^2}'
latex_formula_dimension = r'd = 3'
latex_formula_bounds = r'x_i \in [0, 1], \forall i \in \llbracket 1, d\rrbracket'
latex_formula_global_optimum = r'f([0.11461292, 0.55564907, 0.85254697]) = -3.8627821478'
continuous = True
linear = False
convex = True
unimodal = False
separable = False
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 = 3
self.check_ndim_and_bounds(ndim, bounds, np.array([[0., 1.] for _ in range(self.dim_default)]))
self.f_global = -3.8627821478
self.x_global = np.array([0.11461292, 0.55564907, 0.85254697])
self.a = np.asarray([[3.0, 10., 30.],
[0.1, 10., 35.],
[3.0, 10., 30.],
[0.1, 10., 35.]])
self.p = np.asarray([[0.3689, 0.1170, 0.2673],
[0.4699, 0.4387, 0.7470],
[0.1091, 0.8732, 0.5547],
[0.03815, 0.5743, 0.8828]])
self.c = np.asarray([1., 1.2, 3., 3.2])
[docs] def evaluate(self, x, *args):
self.check_solution(x)
self.n_fe += 1
XX = np.atleast_2d(x)
d = np.sum(self.a * (XX - self.p) ** 2, axis=1)
return -np.sum(self.c * np.exp(-d))
[docs]class Hartmann6(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.
"""
name = "Hartman 6 Function"
latex_formula = r'f(x) = -\sum\limits_{i=1}^{4} c_i e^{-\sum\limits_{j=1}^{n}a_{ij}(x_j - p_{ij})^2}'
latex_formula_dimension = r'd = 3'
latex_formula_bounds = r'x_i \in [0, 1], \forall i \in \llbracket 1, d\rrbracket'
latex_formula_global_optimum = r'f([0.20168952, 0.15001069, 0.47687398, 0.27533243, 0.31165162, 0.65730054]) = -3.32236801141551'
continuous = True
linear = False
convex = True
unimodal = False
separable = False
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 = 6
self.check_ndim_and_bounds(ndim, bounds, np.array([[0., 1.] for _ in range(self.dim_default)]))
self.f_global = -3.32236801141551
self.x_global = np.array([0.20168952, 0.15001069, 0.47687398, 0.27533243, 0.31165162, 0.65730054])
self.a = np.asarray([[10., 3., 17., 3.5, 1.7, 8.],
[0.05, 10., 17., 0.1, 8., 14.],
[3., 3.5, 1.7, 10., 17., 8.],
[17., 8., 0.05, 10., 0.1, 14.]])
self.p = np.asarray([[0.1312, 0.1696, 0.5569, 0.0124, 0.8283, 0.5886],
[0.2329, 0.4135, 0.8307, 0.3736, 0.1004, 0.9991],
[0.2348, 0.1451, 0.3522, 0.2883, 0.3047, 0.665],
[0.4047, 0.8828, 0.8732, 0.5743, 0.1091, 0.0381]])
self.c = np.asarray([1.0, 1.2, 3.0, 3.2])
[docs] def evaluate(self, x, *args):
self.check_solution(x)
self.n_fe += 1
XX = np.atleast_2d(x)
d = np.sum(self.a * (XX - self.p) ** 2, axis=1)
return -np.sum(self.c * np.exp(-d))
[docs]class HelicalValley(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.
"""
name = "Helical Valley"
latex_formula = r'f(x) = 100{[z-10\Psi(x_1,x_2)]^2 +(\sqrt{x_1^2+x_2^2}-1)^2}+x_3^2'
latex_formula_dimension = r'd \in N^+'
latex_formula_bounds = r'x_i \in [-10, 10], \forall i \in \llbracket 1, d\rrbracket'
latex_formula_global_optimum = r'f([1.0, 0.0, 0.0]) = 0'
continuous = True
linear = False
convex = True
unimodal = False
separable = False
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 = 3
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.array([1.0, 0.0, 0.0])
[docs] def evaluate(self, x, *args):
self.check_solution(x)
self.n_fe += 1
r = np.sqrt(x[0] ** 2 + x[1] ** 2)
theta = 1 / (2. * np.pi) * np.arctan2(x[1], x[0])
return x[2] ** 2 + 100 * ((x[2] - 10 * theta) ** 2 + (r - 1) ** 2)
[docs]class Himmelblau(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.
"""
name = "Himmelblau Function"
latex_formula = r'f(x) = (x_1^2 + x_2 - 11)^2 + (x_1 + x_2^2 - 7)^2'
latex_formula_dimension = r'd \in N^+'
latex_formula_bounds = r'x_i \in [-5, 5], \forall i \in \llbracket 1, d\rrbracket'
latex_formula_global_optimum = r'f([3, 2]) = 0'
continuous = True
linear = False
convex = True
unimodal = False
separable = False
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([[-5., 5.] for _ in range(self.dim_default)]))
self.f_global = 0.0
self.x_global = np.array([3., 2.])
[docs] def evaluate(self, x, *args):
self.check_solution(x)
self.n_fe += 1
return (x[0] ** 2 + x[1] - 11) ** 2 + (x[0] + x[1] ** 2 - 7) ** 2
[docs]class Hosaki(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.
"""
name = "Hosaki Function"
latex_formula = r'f(x) = \left ( 1 - 8 x_1 + 7 x_1^2 - \frac{7}{3} x_1^3 + \frac{1}{4} x_1^4 \right ) x_2^2 e^{-x_1}'
latex_formula_dimension = r'd = 2'
latex_formula_bounds = r' 0 <= x_1 <= 5, 0 <= x2 <= 6'
latex_formula_global_optimum = r'f(4, 2) = −2.3458'
continuous = True
linear = False
convex = True
unimodal = False
separable = False
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([[0., 5.], [0., 6.]]))
self.f_global = -2.345811576101292
self.x_global = np.array([4., 2.])
[docs] def evaluate(self, x, *args):
self.check_solution(x)
self.n_fe += 1
val = (1 - 8 * x[0] + 7 * x[0] ** 2 - 7 / 3. * x[0] ** 3 + 0.25 * x[0] ** 4)
return val * x[1] ** 2 * np.exp(-x[1])
[docs]class HolderTable(Benchmark):
"""
.. [1] Gavana, A. Global Optimization Benchmarks and AMPGO retrieved 2015
"""
name = "Hosaki Function"
latex_formula = r'f(x) = - \left|{e^{\left|{1' +\
r'- \frac{\sqrt{x_{1}^{2} + x_{2}^{2}}}{\pi} }\right|} \sin\left(x_{1}\right) \cos\left(x_{2}\right)}\right|'
latex_formula_dimension = r'd = 2'
latex_formula_bounds = r' 0 <= x_1 <= 5, 0 <= x2 <= 6'
latex_formula_global_optimum = r'f(\pm 9.664590028909654) = -19.20850256788675'
continuous = True
linear = False
convex = True
unimodal = False
separable = False
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([[-10., 10.] for _ in range(self.dim_default)]))
self.f_global = -19.20850256788675
self.x_global = np.array([8.055023472141116, 9.664590028909654])
self.x_globals = np.array([[8.055023472141116, 9.664590028909654],
[-8.055023472141116, 9.664590028909654],
[8.055023472141116, -9.664590028909654],
[-8.055023472141116, -9.664590028909654]])
[docs] def evaluate(self, x, *args):
self.check_solution(x)
self.n_fe += 1
return -np.abs(np.sin(x[0]) * np.cos(x[1]) * np.exp(np.abs(1 - np.sqrt(x[0] ** 2 + x[1] ** 2) / np.pi)))