#!/usr/bin/env python
# Created by "Thieu" at 09:33, 21/07/2022 ----------%
# Email: nguyenthieu2102@gmail.com %
# Github: https://github.com/thieu1995 %
# --------------------------------------------------%
import numpy as np
from opfunu.benchmark import Benchmark
[docs]class Damavandi(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 = "Damavandi Function"
latex_formula = r'f(x) = \left[ 1 - \lvert{\frac{\sin[\pi (x_1 - 2)]\sin[\pi (x2 - 2)]}' \
r'{\pi^2 (x_1 - 2)(x_2 - 2)}}\rvert^5 \right] \left[2 + (x_1 - 7)^2 + 2(x_2 - 7)^2 \right]'
latex_formula_dimension = r'd = 2'
latex_formula_bounds = r'x_i \in [0, 14], \forall i \in \llbracket 1, d\rrbracket'
latex_formula_global_optimum = r'f(2, 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([[0., 14.] for _ in range(self.dim_default)]))
self.f_global = 0.0
self.x_global = 2 * np.ones(self.ndim)
[docs] def evaluate(self, x, *args):
self.check_solution(x)
self.n_fe += 1
num = np.sin(np.pi * (x[0] - 2.0)) * np.sin(np.pi * (x[1] - 2.0))
den = (np.pi ** 2) * (x[0] - 2.0) * (x[1] - 2.0) + self.epsilon
factor1 = 1.0 - (np.abs(num / den)) ** 5.0
factor2 = 2 + (x[0] - 7.0) ** 2.0 + 2 * (x[1] - 7.0) ** 2.0
return factor1 * factor2
[docs]class Deb01(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 = "Deb 1 Function"
latex_formula = r'f(x) = - \frac{1}{N} \sum_{i=1}^n \sin^6(5 \pi x_i)'
latex_formula_dimension = r'd = 2'
latex_formula_bounds = r'x_i \in [-1, 1], \forall i \in \llbracket 1, d\rrbracket'
latex_formula_global_optimum = r'f(0.3, -0.3) = -1'
continuous = True
linear = False
convex = True
unimodal = False
separable = True
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 = True
self.dim_default = 2
self.check_ndim_and_bounds(ndim, bounds, np.array([[-1, -1.] for _ in range(self.dim_default)]))
self.f_global = -1.0
self.x_global = np.zeros(self.ndim)
[docs] def evaluate(self, x, *args):
self.check_solution(x)
self.n_fe += 1
return -(1.0 / self.ndim) * np.sum(np.sin(5 * np.pi * x) ** 6.0)
[docs]class Deb03(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 = "Deb 3 Function"
latex_formula = r'f(x) = - \frac{1}{N} \sum_{i=1}^n \sin^6 \left[ 5 \pi\left ( x_i^{3/4} - 0.05 \right) \right ]'
latex_formula_dimension = r'd = 2'
latex_formula_bounds = r'x_i \in [-1, 1], \forall i \in \llbracket 1, d\rrbracket'
latex_formula_global_optimum = r'f(0.3, -0.3) = -1'
continuous = True
linear = False
convex = True
unimodal = False
separable = True
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 = True
self.dim_default = 2
self.check_ndim_and_bounds(ndim, bounds, np.array([[-1., 1.] for _ in range(self.dim_default)]))
self.f_global = -1.0
self.x_global = np.zeros(self.ndim)
[docs] def evaluate(self, x, *args):
self.check_solution(x)
self.n_fe += 1
return -(1.0 / self.ndim) * np.sum(np.sin(5 * np.pi * (x ** 0.75 - 0.05)) ** 6.0)
[docs]class Decanomial(Benchmark):
"""
.. [1] Gavana, A. Global Optimization Benchmarks and AMPGO retrieved 2015
"""
name = "Decanomial Function"
latex_formula = r'f(x) = 0.001 \left(\lvert{x_{2}^{4} + 12 x_{2}^{3}' + \
r'+ 54 x_{2}^{2} + 108 x_{2} + 81.0}\rvert + \lvert{x_{1}^{10} - 20 x_{1}^{9} + 180 x_{1}^{8} - 960 x_{1}^{7} + 3360 x_{1}^{6}' + \
r'- 8064 x_{1}^{5} + 13340 x_{1}^{4} - 15360 x_{1}^{3} + 11520 x_{1}^{2} - 5120 x_{1} + 2624.0}\rvert\right)^{2}'
latex_formula_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(2, -3) = 0'
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 = 0.
self.x_global = np.array([2, -3])
[docs] def evaluate(self, x, *args):
self.check_solution(x)
self.n_fe += 1
val = x[1] ** 4 + 12 * x[1] ** 3 + 54 * x[1] ** 2 + 108 * x[1] + 81.0
val2 = x[0] ** 10. - 20 * x[0] ** 9 + 180 * x[0] ** 8 - 960 * x[0] ** 7
val2 += 3360 * x[0] ** 6 - 8064 * x[0] ** 5 + 13340 * x[0] ** 4
val2 += - 15360 * x[0] ** 3 + 11520 * x[0] ** 2 - 5120 * x[0] + 2624
return 0.001 * (abs(val) + abs(val2)) ** 2.
[docs]class Deceptive(Benchmark):
"""
.. [1] Gavana, A. Global Optimization Benchmarks and AMPGO retrieved 2015
"""
name = "Deceptive Function"
latex_formula = r'f(x) = - \left [\frac{1}{n} \sum_{i=1}^{n} g_i(x_i) \right ]^{\beta}'
latex_formula_dimension = r'd \in N^+'
latex_formula_bounds = r'x_i \in [0, 1], \forall i \in \llbracket 1, d\rrbracket'
latex_formula_global_optimum = r'f(alpha_i) = -1'
continuous = True
linear = False
convex = False
unimodal = False
separable = False
differentiable = True
scalable = True
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 = True
self.dim_default = 2
self.check_ndim_and_bounds(ndim, bounds, np.array([[0., 1.] for _ in range(self.dim_default)]))
self.f_global = -1.0
self.x_global = np.arange(1.0, self.ndim + 1.0) / (self.ndim + 1.0)
[docs] def evaluate(self, x, *args):
self.check_solution(x)
self.n_fe += 1
alpha = np.arange(1.0, self.ndim + 1.0) / (self.ndim + 1.0)
beta = 2.0
g = np.zeros((self.ndim,))
for i in range(self.ndim):
if x[i] <= 0.0:
g[i] = x[i]
elif x[i] < 0.8 * alpha[i]:
g[i] = -x[i] / alpha[i] + 0.8
elif x[i] < alpha[i]:
g[i] = 5.0 * x[i] / alpha[i] - 4.0
elif x[i] < (1.0 + 4 * alpha[i]) / 5.0:
g[i] = 5.0 * (x[i] - alpha[i]) / (alpha[i] - 1.0) + 1.0
elif x[i] <= 1.0:
g[i] = (x[i] - 1.0) / (1.0 - alpha[i]) + 4.0 / 5.0
else:
g[i] = x[i] - 1.0
return -((1.0 / self.ndim) * np.sum(g)) ** beta
[docs]class DeckkersAarts(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 = "Deckkers-Aarts Function"
latex_formula = r'f(x) = 10^5x_1^2 + x_2^2 - (x_1^2 + x_2^2)^2 + 10^{-5}(x_1^2 + x_2^2)^4'
latex_formula_dimension = r'd = 2'
latex_formula_bounds = r'x_i \in [-20, 20], \forall i \in \llbracket 1, d\rrbracket'
latex_formula_global_optimum = r'f(0, \pm 14.9451209) = -24776.518242168'
continuous = True
linear = False
convex = False
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([[-20., 20.] for _ in range(self.dim_default)]))
self.f_global = -24776.5183421814
self.x_global = np.array([0, 14.9451209])
self.x_globals = np.array([[0, 14.9451209], [0, -14.9451209]])
[docs] def evaluate(self, x, *args):
self.check_solution(x)
self.n_fe += 1
return (1.e5 * x[0] ** 2 + x[1] ** 2 - (x[0] ** 2 + x[1] ** 2) ** 2 + 1.e-5 * (x[0] ** 2 + x[1] ** 2) ** 4)
[docs]class DeflectedCorrugatedSpring(Benchmark):
"""
.. [1] Gavana, A. Global Optimization Benchmarks and AMPGO retrieved 2015
"""
name = "Deflected Corrugated Spring Function"
latex_formula = r'f(x) = 0.1\sum_{i=1}^n \left[ (x_i - \alpha)^2 - \cos \left( K \sqrt {\sum_{i=1}^n (x_i - \alpha)^2}\right ) \right ]'
latex_formula_dimension = r'd = 2'
latex_formula_bounds = r'x_i \in [-0, 2\alpha], \forall i \in \llbracket 1, d\rrbracket'
latex_formula_global_optimum = r'f(x_i) = f(alpha_i) = -1'
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, alpha=5.0):
super().__init__()
self.dim_changeable = True
self.dim_default = 2
self.alpha = alpha
self.check_ndim_and_bounds(ndim, bounds, np.array([[0., 2.*self.alpha] for _ in range(self.dim_default)]))
self.f_global = -1.0
self.x_global = self.alpha * np.ones(self.ndim)
[docs] def evaluate(self, x, *args):
self.check_solution(x)
self.n_fe += 1
K = 5.0
return -np.cos(K * np.sqrt(np.sum((x - self.alpha) ** 2))) + 0.1 * np.sum((x - self.alpha) ** 2)
[docs]class DeVilliersGlasser01(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 = "DeVilliers-Glasser 1 Function"
latex_formula = r'f(x) = \sum_{i=1}^{24} \left[ x_1x_2^{t_i}\sin(x_3t_i + x_4) - y_i \right ]^2'
latex_formula_dimension = r'd = 4'
latex_formula_bounds = r'x_i \in [-500, 500], \forall i \in \llbracket 1, d\rrbracket'
latex_formula_global_optimum = r'f(60.137, 1.371, 3.112, 1.761) = 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 = 4
self.check_ndim_and_bounds(ndim, bounds, np.array([[-500., 500.] for _ in range(self.dim_default)]))
self.f_global = 0.0
self.x_global = np.array([60.137, 1.371, 3.112, 1.761])
[docs] def evaluate(self, x, *args):
self.check_solution(x)
self.n_fe += 1
t = 0.1 * np.arange(24)
y = 60.137 * (1.371 ** t) * np.sin(3.112 * t + 1.761)
return sum((x[0] * (x[1] ** t) * np.sin(x[2] * t + x[3]) - y) ** 2.0)
[docs]class DeVilliersGlasser02(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 = "DeVilliers-Glasser 2 Function"
latex_formula = r'f(x) = \sum_{i=1}^{24} \left[ x_1x_2^{t_i}' +\
r'\tanh \left [x_3t_i + \sin(x_4t_i) \right] \cos(t_ie^{x_5}) - y_i \right ]^2'
latex_formula_dimension = r'd = 5'
latex_formula_bounds = r'x_i \in [-500, 500], \forall i \in \llbracket 1, d\rrbracket'
latex_formula_global_optimum = r'f(53.81, 1.27, 3.012, 2.13, 0.507) = 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 = 5
self.check_ndim_and_bounds(ndim, bounds, np.array([[-500., 500.] for _ in range(self.dim_default)]))
self.f_global = 0.0
self.x_global = np.array([53.81, 1.27, 3.012, 2.13, 0.507])
[docs] def evaluate(self, x, *args):
self.check_solution(x)
self.n_fe += 1
t = 0.1 * np.arange(16)
y = (53.81 * 1.27 ** t * np.tanh(3.012 * t + np.sin(2.13 * t)) * np.cos(np.exp(0.507) * t))
return np.sum((x[0] * (x[1] ** t) * np.tanh(x[2] * t + np.sin(x[3] * t)) * np.cos(t * np.exp(x[4])) - y) ** 2.0)
[docs]class DixonPrice(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 = "Dixon & Price Function"
latex_formula = r'f(x) = (x_i - 1)^2 + \sum_{i=2}^n i(2x_i^2 - x_{i-1})^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(2^{- \frac{(2^i - 2)}{2^i}}) = 0'
continuous = True
linear = False
convex = True
unimodal = True
separable = False
differentiable = True
scalable = True
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 = 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.array([2.0 ** (-(2.0 ** i - 2.0) / 2.0 ** i) for i in range(1, self.ndim + 1)])
[docs] def evaluate(self, x, *args):
self.check_solution(x)
self.n_fe += 1
i = np.arange(2, self.ndim + 1)
s = i * (2.0 * x[1:] ** 2.0 - x[:-1]) ** 2.0
return np.sum(s) + (x[0] - 1.0) ** 2.0
[docs]class Dolan(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 = "Dolan Function"
latex_formula = r'f(x) = \lvert (x_1 + 1.7 x_2)\sin(x_1) - 1.5 x_3 - 0.1 x_4\cos(x_5 + x_5 - x_1) + 0.2 x_5^2 - x_2 - 1 \rvert'
latex_formula_dimension = r'd = 5'
latex_formula_bounds = r'x_i \in [-100, 100], \forall i \in \llbracket 1, d\rrbracket'
latex_formula_global_optimum = r'f(8.39045925, 4.81424707, 7.34574133, 68.88246895, 3.85470806) = 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 = 5
self.check_ndim_and_bounds(ndim, bounds, np.array([[-100., 100.] for _ in range(self.dim_default)]))
self.f_global = 0.0
self.x_global = np.array([-74.10522498, 44.33511286, 6.21069214, 18.42772233, -16.5839403])
[docs] def evaluate(self, x, *args):
self.check_solution(x)
self.n_fe += 1
return (abs((x[0] + 1.7 * x[1]) * np.sin(x[0]) - 1.5 * x[2]
- 0.1 * x[3] * np.cos(x[3] + x[4] - x[0]) + 0.2 * x[4] ** 2 - x[1] - 1))
[docs]class DropWave(Benchmark):
"""
.. [1] Gavana, A. Global Optimization Benchmarks and AMPGO retrieved 2015
"""
name = "DropWave Function"
latex_formula = r'f(x) = - \frac{1 + \cos\left(12 \sqrt{\sum_{i=1}^{n} x_i^{2}}\right)}{2 + 0.5 \sum_{i=1}^{n} x_i^{2}}'
latex_formula_dimension = r'd = 2'
latex_formula_bounds = r'x_i \in [-5.12, 5.12], \forall i \in \llbracket 1, d\rrbracket'
latex_formula_global_optimum = r'f(0, 0) = -1'
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 = True
self.dim_default = 2
self.check_ndim_and_bounds(ndim, bounds, np.array([[-5.12, 5.12] for _ in range(self.dim_default)]))
self.f_global = -1.0
self.x_global = np.array([0., 0.])
[docs] def evaluate(self, x, *args):
self.check_solution(x)
self.n_fe += 1
norm_x = np.sum(x ** 2)
return -(1 + np.cos(12 * np.sqrt(norm_x))) / (0.5 * norm_x + 2)