#!/usr/bin/env python
# Created by "Thieu" at 17:32, 30/07/2022 ----------%
# Email: nguyenthieu2102@gmail.com %
# Github: https://github.com/thieu1995 %
# --------------------------------------------------%
import numpy as np
from opfunu.benchmark import Benchmark
[docs]class Zacharov(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{Zacharov}}(x) = \sum_{i=1}^{n} x_i^2 + \left ( \frac{1}{2}\sum_{i=1}^{n} i x_i \right )^2
+ \left ( \frac{1}{2} \sum_{i=1}^{n} i x_i \right )^4
Here :math:`x_i \in [-5, 10]` for :math:`i = 1, ..., n`.
*Global optimum*: :math:`f(x) = 0.0`for :math:`x = [0, 0,,,0]`
"""
name = "Zacharov Function"
latex_formula = "\sum_{i=1}^{n} x_i^2 + \left ( \frac{1}{2}\sum_{i=1}^{n} i x_i \right )^2+ \left ( \frac{1}{2} \sum_{i=1}^{n} i x_i \right )^4"
latex_formula_dimension = r'd = n'
latex_formula_bounds = r'x_i \in [-5, 10]'
latex_formula_global_optimum = r'f(0, 0,...,0) = 0'
continuous = False
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., 10.] for _ in range(self.dim_default)]))
self.f_global = 0.0
self.x_global = np.zeros(self.ndim)
[docs] def evaluate(self, x, *args):
self.check_solution(x)
self.n_fe += 1
u = np.sum(x ** 2)
v = np.sum(np.arange(1, self.ndim + 1) * x)
return u + (0.5 * v) ** 2 + (0.5 * v) ** 4
[docs]class ZeroSum(Benchmark):
"""
.. [1] Gavana, A. Global Optimization Benchmarks and AMPGO retrieved 2015
.. math::
f_{\text{ZeroSum}}(x) = \begin{cases} 0 & \textrm{if} \sum_{i=1}^n x_i = 0 \\
1 + \left(10000 \left |\sum_{i=1}^n x_i\right| \right)^{0.5} & \textrm{otherwise} \end{cases}
Here :math:`x_i \in [-10, 10]` for :math:`i = 1, ..., n`.
*Global optimum*: :math:`f(x) = 0.0`for :math:`\sum_{i=1}^n x_i = 0`
"""
name = "ZeroSum Function"
latex_formula = "\begin{cases} 0 & \textrm{if} \sum_{i=1}^n x_i = 0 \\ 1 + \left(10000 \left |\sum_{i=1}^n x_i\right| \right)^{0.5} & \textrm{otherwise} \end{cases}"
latex_formula_dimension = r'd = n'
latex_formula_bounds = r'x_i \in [-10, 10]'
latex_formula_global_optimum = r'f(x_best) = 0'
continuous = False
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([[-10., 10.] for _ in range(self.dim_default)]))
self.f_global = 0.0
self.x_global = np.zeros(self.ndim)
[docs] def evaluate(self, x, *args):
self.check_solution(x)
self.n_fe += 1
if np.abs(np.sum(x)) < 3e-16:
return 0.0
return 1.0 + (10000.0 * np.abs(np.sum(x))) ** 0.5
[docs]class Zettl(Benchmark):
"""
.. [1] Gavana, A. Global Optimization Benchmarks and AMPGO retrieved 2015
.. math::
f_{\text{Zettl}}(x) = \frac{1}{4} x_{1} + \left(x_{1}^{2} - 2 x_{1}
+ x_{2}^{2}\right)^{2}
Here :math:`x_i \in [-1, 5]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x) = -0.0037912`for :math:`x = [-0.029896, 0.0]`
"""
name = "Zettl Function"
latex_formula = "\sum_{i=1}^{n} x_i^2 + \left ( \frac{1}{2}\sum_{i=1}^{n} i x_i \right )^2+ \left ( \frac{1}{2} \sum_{i=1}^{n} i x_i \right )^4"
latex_formula_dimension = r'd = 2'
latex_formula_bounds = r'x_i \in [-0.029896, 0.0]'
latex_formula_global_optimum = r'f(x_best) = -0.0037912'
continuous = False
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([[-1., 5.] for _ in range(self.dim_default)]))
self.f_global = -0.0037912
self.x_global = np.array([-0.029896, 0.0])
[docs] def evaluate(self, x, *args):
self.check_solution(x)
self.n_fe += 1
return (x[0] ** 2 + x[1] ** 2 - 2 * x[0]) ** 2 + 0.25 * x[0]
[docs]class Zimmerman(Benchmark):
"""
.. [1] Gavana, A. Global Optimization Benchmarks and AMPGO retrieved 2015
.. math::
f_{\text{Zimmerman}}(x) = \max \left[Zh1(x), Zp(Zh2(x))\textrm{sgn}(Zh2(x)), Zp(Zh3(x))\textrm{sgn}(Zh3(x)),
Zp(-x_1)\textrm{sgn}(x_1),Zp(-x_2)\textrm{sgn}(x_2) \right]
.. math::
\begin{cases}
Zh1(x) = 9 - x_1 - x_2 \\
Zh2(x) = (x_1 - 3)^2 + (x_2 - 2)^2 \\
Zh3(x) = x_1x_2 - 14 \\
Zp(t) = 100(1 + t)
\end{cases}
Where :math:`x` is a vector and :math:`t` is a scalar.
Here, :math:`x_i \in [0, 100]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x) = 0` for :math:`x = [7, 2]`
"""
name = "Zimmerman Function"
latex_formula = "\max \left[Zh1(x), Zp(Zh2(x))\textrm{sgn}(Zh2(x)), Zp(Zh3(x))\textrm{sgn}(Zh3(x)), " \
"Zp(-x_1)\textrm{sgn}(x_1),Zp(-x_2)\textrm{sgn}(x_2) \right]"
latex_formula_dimension = r'd = 2'
latex_formula_bounds = r'x_i \in [0, 100]'
latex_formula_global_optimum = r'f([7, 2]) = 0.'
continuous = False
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, 100] for _ in range(self.dim_default)]))
self.f_global = 0.
self.x_global = np.array([7.0, 2.0])
[docs] def evaluate(self, x, *args):
self.check_solution(x)
self.n_fe += 1
Zh1 = lambda x: 9.0 - x[0] - x[1]
Zh2 = lambda x: (x[0] - 3.0) ** 2.0 + (x[1] - 2.0) ** 2.0 - 16.0
Zh3 = lambda x: x[0] * x[1] - 14.0
Zp = lambda x: 100.0 * (1.0 + x)
return max(Zh1(x), Zp(Zh2(x)) * np.sign(Zh2(x)), Zp(Zh3(x)) * np.sign(Zh3(x)), Zp(-x[0]) * np.sign(x[0]),Zp(-x[1]) * np.sign(x[1]))
[docs]class Zirilli(Benchmark):
"""
.. [1] Gavana, A. Global Optimization Benchmarks and AMPGO retrieved 2015
.. math::
f_{\text{Zirilli}}(x) = 0.25x_1^4 - 0.5x_1^2 + 0.1x_1 + 0.5x_2^2
.. math::
\begin{cases}
Zh1(x) = 9 - x_1 - x_2 \\
Zh2(x) = (x_1 - 3)^2 + (x_2 - 2)^2 \\
Zh3(x) = x_1x_2 - 14 \\
Zp(t) = 100(1 + t)
\end{cases}
Where :math:`x` is a vector and :math:`t` is a scalar.
Here, :math:`x_i \in [-10, 10]` for :math:`i = 1, 2`.
*Global optimum*: :math:`f(x) = -0.3523` for :math:`x = [-1.0465, 0]`
"""
name = "Zirilli Function"
latex_formula = "f_{\text{Zirilli}}(x) = 0.25x_1^4 - 0.5x_1^2 + 0.1x_1 + 0.5x_2^2"
latex_formula_dimension = r'd = 2'
latex_formula_bounds = r'x_i \in [-10, 10]'
latex_formula_global_optimum = r'f([-1.0465, 0]) = -0.3523'
continuous = False
linear = False
convex = True
unimodal = True
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 = -0.3523
self.x_global = np.array([-1.0465, 0])
[docs] def evaluate(self, x, *args):
self.check_solution(x)
self.n_fe += 1
return 0.25 * x[0] ** 4 - 0.5 * x[0] ** 2 + 0.1 * x[0] + 0.5 * x[1] ** 2