727 lines
20 KiB
Python
727 lines
20 KiB
Python
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# -*- coding: utf-8 -*-
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from numpy import (abs, sum, sin, cos, sqrt, log, prod, where, pi, exp, arange,
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floor, log10, atleast_2d, zeros)
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from .go_benchmark import Benchmark
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class Parsopoulos(Benchmark):
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r"""
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Parsopoulos objective function.
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This class defines the Parsopoulos [1]_ global optimization problem. This is a
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multimodal minimization problem defined as follows:
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.. math::
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f_{\text{Parsopoulos}}(x) = \cos(x_1)^2 + \sin(x_2)^2
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with :math:`x_i \in [-5, 5]` for :math:`i = 1, 2`.
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*Global optimum*: This function has infinite number of global minima in R2,
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at points :math:`\left(k\frac{\pi}{2}, \lambda \pi \right)`,
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where :math:`k = \pm1, \pm3, ...` and :math:`\lambda = 0, \pm1, \pm2, ...`
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In the given domain problem, function has 12 global minima all equal to
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zero.
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.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
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For Global Optimization Problems Int. Journal of Mathematical Modelling
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and Numerical Optimisation, 2013, 4, 150-194.
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"""
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def __init__(self, dimensions=2):
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Benchmark.__init__(self, dimensions)
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self._bounds = list(zip([-5.0] * self.N, [5.0] * self.N))
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self.global_optimum = [[pi / 2.0, pi]]
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self.fglob = 0
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def fun(self, x, *args):
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self.nfev += 1
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return cos(x[0]) ** 2.0 + sin(x[1]) ** 2.0
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class Pathological(Benchmark):
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r"""
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Pathological objective function.
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This class defines the Pathological [1]_ global optimization problem. This is a
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multimodal minimization problem defined as follows:
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.. math::
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f_{\text{Pathological}}(x) = \sum_{i=1}^{n -1} \frac{\sin^{2}\left(
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\sqrt{100 x_{i+1}^{2} + x_{i}^{2}}\right) -0.5}{0.001 \left(x_{i}^{2}
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- 2x_{i}x_{i+1} + x_{i+1}^{2}\right)^{2} + 0.50}
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Here, :math:`n` represents the number of dimensions and
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:math:`x_i \in [-100, 100]` for :math:`i = 1, 2`.
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*Global optimum*: :math:`f(x) = 0.` for :math:`x = [0, 0]` for
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:math:`i = 1, 2`
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.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
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For Global Optimization Problems Int. Journal of Mathematical Modelling
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and Numerical Optimisation, 2013, 4, 150-194.
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"""
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def __init__(self, dimensions=2):
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Benchmark.__init__(self, dimensions)
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self._bounds = list(zip([-100.0] * self.N,
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[100.0] * self.N))
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self.global_optimum = [[0 for _ in range(self.N)]]
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self.fglob = 0.
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def fun(self, x, *args):
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self.nfev += 1
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vec = (0.5 + (sin(sqrt(100 * x[: -1] ** 2 + x[1:] ** 2)) ** 2 - 0.5) /
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(1. + 0.001 * (x[: -1] ** 2 - 2 * x[: -1] * x[1:]
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+ x[1:] ** 2) ** 2))
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return sum(vec)
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class Paviani(Benchmark):
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r"""
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Paviani objective function.
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This class defines the Paviani [1]_ global optimization problem. This is a
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multimodal minimization problem defined as follows:
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.. math::
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f_{\text{Paviani}}(x) = \sum_{i=1}^{10} \left[\log^{2}\left(10
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- x_i\right) + \log^{2}\left(x_i -2\right)\right]
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- \left(\prod_{i=1}^{10} x_i^{10} \right)^{0.2}
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with :math:`x_i \in [2.001, 9.999]` for :math:`i = 1, ... , 10`.
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*Global optimum*: :math:`f(x_i) = -45.7784684040686` for
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:math:`x_i = 9.350266` for :math:`i = 1, ..., 10`
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.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
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For Global Optimization Problems Int. Journal of Mathematical Modelling
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and Numerical Optimisation, 2013, 4, 150-194.
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TODO: think Gavana web/code definition is wrong because final product term
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shouldn't raise x to power 10.
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"""
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def __init__(self, dimensions=10):
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Benchmark.__init__(self, dimensions)
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self._bounds = list(zip([2.001] * self.N, [9.999] * self.N))
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self.global_optimum = [[9.350266 for _ in range(self.N)]]
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self.fglob = -45.7784684040686
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def fun(self, x, *args):
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self.nfev += 1
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return sum(log(x - 2) ** 2.0 + log(10.0 - x) ** 2.0) - prod(x) ** 0.2
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class Penalty01(Benchmark):
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r"""
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Penalty 1 objective function.
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This class defines the Penalty 1 [1]_ global optimization problem. This is a
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imultimodal minimization problem defined as follows:
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.. math::
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f_{\text{Penalty01}}(x) = \frac{\pi}{30} \left\{10 \sin^2(\pi y_1)
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+ \sum_{i=1}^{n-1} (y_i - 1)^2 \left[1 + 10 \sin^2(\pi y_{i+1}) \right]
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+ (y_n - 1)^2 \right \} + \sum_{i=1}^n u(x_i, 10, 100, 4)
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Where, in this exercise:
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.. math::
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y_i = 1 + \frac{1}{4}(x_i + 1)
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And:
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.. math::
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u(x_i, a, k, m) =
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\begin{cases}
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k(x_i - a)^m & \textrm{if} \hspace{5pt} x_i > a \\
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0 & \textrm{if} \hspace{5pt} -a \leq x_i \leq a \\
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k(-x_i - a)^m & \textrm{if} \hspace{5pt} x_i < -a
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\end{cases}
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Here, :math:`n` represents the number of dimensions and
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:math:`x_i \in [-50, 50]` for :math:`i= 1, ..., n`.
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*Global optimum*: :math:`f(x) = 0` for :math:`x_i = -1` for
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:math:`i = 1, ..., n`
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.. [1] Gavana, A. Global Optimization Benchmarks and AMPGO retrieved 2015
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"""
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def __init__(self, dimensions=2):
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Benchmark.__init__(self, dimensions)
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self._bounds = list(zip([-50.0] * self.N, [50.0] * self.N))
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self.custom_bounds = ([-5.0, 5.0], [-5.0, 5.0])
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self.global_optimum = [[-1.0 for _ in range(self.N)]]
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self.fglob = 0.0
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self.change_dimensionality = True
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def fun(self, x, *args):
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self.nfev += 1
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a, b, c = 10.0, 100.0, 4.0
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xx = abs(x)
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u = where(xx > a, b * (xx - a) ** c, 0.0)
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y = 1.0 + (x + 1.0) / 4.0
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return (sum(u) + (pi / 30.0) * (10.0 * sin(pi * y[0]) ** 2.0
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+ sum((y[: -1] - 1.0) ** 2.0
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* (1.0 + 10.0 * sin(pi * y[1:]) ** 2.0))
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+ (y[-1] - 1) ** 2.0))
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class Penalty02(Benchmark):
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r"""
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Penalty 2 objective function.
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This class defines the Penalty 2 [1]_ global optimization problem. This is a
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multimodal minimization problem defined as follows:
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.. math::
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f_{\text{Penalty02}}(x) = 0.1 \left\{\sin^2(3\pi x_1) + \sum_{i=1}^{n-1}
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(x_i - 1)^2 \left[1 + \sin^2(3\pi x_{i+1}) \right ]
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+ (x_n - 1)^2 \left [1 + \sin^2(2 \pi x_n) \right ]\right \}
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+ \sum_{i=1}^n u(x_i, 5, 100, 4)
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Where, in this exercise:
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.. math::
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u(x_i, a, k, m) =
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\begin{cases}
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k(x_i - a)^m & \textrm{if} \hspace{5pt} x_i > a \\
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0 & \textrm{if} \hspace{5pt} -a \leq x_i \leq a \\
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k(-x_i - a)^m & \textrm{if} \hspace{5pt} x_i < -a \\
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\end{cases}
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Here, :math:`n` represents the number of dimensions and
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:math:`x_i \in [-50, 50]` for :math:`i = 1, ..., n`.
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*Global optimum*: :math:`f(x) = 0` for :math:`x_i = 1` for
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:math:`i = 1, ..., n`
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.. [1] Gavana, A. Global Optimization Benchmarks and AMPGO retrieved 2015
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"""
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def __init__(self, dimensions=2):
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Benchmark.__init__(self, dimensions)
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self._bounds = list(zip([-50.0] * self.N, [50.0] * self.N))
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self.custom_bounds = ([-4.0, 4.0], [-4.0, 4.0])
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self.global_optimum = [[1.0 for _ in range(self.N)]]
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self.fglob = 0.0
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self.change_dimensionality = True
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def fun(self, x, *args):
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self.nfev += 1
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a, b, c = 5.0, 100.0, 4.0
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xx = abs(x)
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u = where(xx > a, b * (xx - a) ** c, 0.0)
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return (sum(u) + 0.1 * (10 * sin(3.0 * pi * x[0]) ** 2.0
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+ sum((x[:-1] - 1.0) ** 2.0
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* (1.0 + sin(3 * pi * x[1:]) ** 2.0))
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+ (x[-1] - 1) ** 2.0 * (1 + sin(2 * pi * x[-1]) ** 2.0)))
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class PenHolder(Benchmark):
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r"""
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PenHolder objective function.
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This class defines the PenHolder [1]_ global optimization problem. This is a
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multimodal minimization problem defined as follows:
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.. math::
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f_{\text{PenHolder}}(x) = -e^{\left|{e^{-\left|{- \frac{\sqrt{x_{1}^{2}
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+ x_{2}^{2}}}{\pi} + 1}\right|} \cos\left(x_{1}\right)
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\cos\left(x_{2}\right)}\right|^{-1}}
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with :math:`x_i \in [-11, 11]` for :math:`i = 1, 2`.
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*Global optimum*: :math:`f(x_i) = -0.9635348327265058` for
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:math:`x_i = \pm 9.646167671043401` for :math:`i = 1, 2`
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.. [1] Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions
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For Global Optimization Problems Int. Journal of Mathematical Modelling
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and Numerical Optimisation, 2013, 4, 150-194.
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"""
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def __init__(self, dimensions=2):
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Benchmark.__init__(self, dimensions)
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self._bounds = list(zip([-11.0] * self.N, [11.0] * self.N))
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self.global_optimum = [[-9.646167708023526, 9.646167671043401]]
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self.fglob = -0.9635348327265058
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def fun(self, x, *args):
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self.nfev += 1
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a = abs(1. - (sqrt(x[0] ** 2 + x[1] ** 2) / pi))
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b = cos(x[0]) * cos(x[1]) * exp(a)
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return -exp(-abs(b) ** -1)
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class PermFunction01(Benchmark):
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r"""
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PermFunction 1 objective function.
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This class defines the PermFunction1 [1]_ global optimization problem. This is
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a multimodal minimization problem defined as follows:
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.. math::
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f_{\text{PermFunction01}}(x) = \sum_{k=1}^n \left\{ \sum_{j=1}^n (j^k
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+ \beta) \left[ \left(\frac{x_j}{j}\right)^k - 1 \right] \right\}^2
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Here, :math:`n` represents the number of dimensions and
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:math:`x_i \in [-n, n + 1]` for :math:`i = 1, ..., n`.
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*Global optimum*: :math:`f(x) = 0` for :math:`x_i = i` for
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:math:`i = 1, ..., n`
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.. [1] Mishra, S. Global Optimization by Differential Evolution and
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Particle Swarm Methods: Evaluation on Some Benchmark Functions.
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Munich Personal RePEc Archive, 2006, 1005
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TODO: line 560
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"""
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def __init__(self, dimensions=2):
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Benchmark.__init__(self, dimensions)
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self._bounds = list(zip([-self.N] * self.N,
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[self.N + 1] * self.N))
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self.global_optimum = [list(range(1, self.N + 1))]
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self.fglob = 0.0
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self.change_dimensionality = True
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def fun(self, x, *args):
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self.nfev += 1
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b = 0.5
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k = atleast_2d(arange(self.N) + 1).T
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j = atleast_2d(arange(self.N) + 1)
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s = (j ** k + b) * ((x / j) ** k - 1)
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return sum((sum(s, axis=1) ** 2))
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class PermFunction02(Benchmark):
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r"""
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PermFunction 2 objective function.
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This class defines the Perm Function 2 [1]_ global optimization problem. This is
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a multimodal minimization problem defined as follows:
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||
|
|
|
||
|
|
.. math::
|
||
|
|
|
||
|
|
f_{\text{PermFunction02}}(x) = \sum_{k=1}^n \left\{ \sum_{j=1}^n (j
|
||
|
|
+ \beta) \left[ \left(x_j^k - {\frac{1}{j}}^{k} \right )
|
||
|
|
\right] \right\}^2
|
||
|
|
|
||
|
|
|
||
|
|
Here, :math:`n` represents the number of dimensions and
|
||
|
|
:math:`x_i \in [-n, n+1]` for :math:`i = 1, ..., n`.
|
||
|
|
|
||
|
|
*Global optimum*: :math:`f(x) = 0` for :math:`x_i = \frac{1}{i}`
|
||
|
|
for :math:`i = 1, ..., n`
|
||
|
|
|
||
|
|
.. [1] Mishra, S. Global Optimization by Differential Evolution and
|
||
|
|
Particle Swarm Methods: Evaluation on Some Benchmark Functions.
|
||
|
|
Munich Personal RePEc Archive, 2006, 1005
|
||
|
|
|
||
|
|
TODO: line 582
|
||
|
|
"""
|
||
|
|
|
||
|
|
def __init__(self, dimensions=2):
|
||
|
|
Benchmark.__init__(self, dimensions)
|
||
|
|
|
||
|
|
self._bounds = list(zip([-self.N] * self.N,
|
||
|
|
[self.N + 1] * self.N))
|
||
|
|
self.custom_bounds = ([0, 1.5], [0, 1.0])
|
||
|
|
|
||
|
|
self.global_optimum = [1. / arange(1, self.N + 1)]
|
||
|
|
self.fglob = 0.0
|
||
|
|
self.change_dimensionality = True
|
||
|
|
|
||
|
|
def fun(self, x, *args):
|
||
|
|
self.nfev += 1
|
||
|
|
|
||
|
|
b = 10
|
||
|
|
k = atleast_2d(arange(self.N) + 1).T
|
||
|
|
j = atleast_2d(arange(self.N) + 1)
|
||
|
|
s = (j + b) * (x ** k - (1. / j) ** k)
|
||
|
|
return sum((sum(s, axis=1) ** 2))
|
||
|
|
|
||
|
|
|
||
|
|
class Pinter(Benchmark):
|
||
|
|
|
||
|
|
r"""
|
||
|
|
Pinter objective function.
|
||
|
|
|
||
|
|
This class defines the Pinter [1]_ global optimization problem. This is a
|
||
|
|
multimodal minimization problem defined as follows:
|
||
|
|
|
||
|
|
.. math::
|
||
|
|
|
||
|
|
f_{\text{Pinter}}(x) = \sum_{i=1}^n ix_i^2 + \sum_{i=1}^n 20i
|
||
|
|
\sin^2 A + \sum_{i=1}^n i \log_{10} (1 + iB^2)
|
||
|
|
|
||
|
|
|
||
|
|
Where, in this exercise:
|
||
|
|
|
||
|
|
.. math::
|
||
|
|
|
||
|
|
\begin{cases}
|
||
|
|
A = x_{i-1} \sin x_i + \sin x_{i+1} \\
|
||
|
|
B = x_{i-1}^2 - 2x_i + 3x_{i + 1} - \cos x_i + 1\\
|
||
|
|
\end{cases}
|
||
|
|
|
||
|
|
Where :math:`x_0 = x_n` and :math:`x_{n + 1} = x_1`.
|
||
|
|
|
||
|
|
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) = 0` for :math:`x_i = 0` for
|
||
|
|
:math:`i = 1, ..., n`
|
||
|
|
|
||
|
|
.. [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.
|
||
|
|
"""
|
||
|
|
|
||
|
|
def __init__(self, dimensions=2):
|
||
|
|
Benchmark.__init__(self, dimensions)
|
||
|
|
|
||
|
|
self._bounds = list(zip([-10.0] * self.N, [10.0] * self.N))
|
||
|
|
|
||
|
|
self.global_optimum = [[0.0 for _ in range(self.N)]]
|
||
|
|
self.fglob = 0.0
|
||
|
|
self.change_dimensionality = True
|
||
|
|
|
||
|
|
def fun(self, x, *args):
|
||
|
|
self.nfev += 1
|
||
|
|
i = arange(self.N) + 1
|
||
|
|
xx = zeros(self.N + 2)
|
||
|
|
xx[1: - 1] = x
|
||
|
|
xx[0] = x[-1]
|
||
|
|
xx[-1] = x[0]
|
||
|
|
A = xx[0: -2] * sin(xx[1: - 1]) + sin(xx[2:])
|
||
|
|
B = xx[0: -2] ** 2 - 2 * xx[1: - 1] + 3 * xx[2:] - cos(xx[1: - 1]) + 1
|
||
|
|
return (sum(i * x ** 2)
|
||
|
|
+ sum(20 * i * sin(A) ** 2)
|
||
|
|
+ sum(i * log10(1 + i * B ** 2)))
|
||
|
|
|
||
|
|
|
||
|
|
class Plateau(Benchmark):
|
||
|
|
|
||
|
|
r"""
|
||
|
|
Plateau objective function.
|
||
|
|
|
||
|
|
This class defines the Plateau [1]_ global optimization problem. This
|
||
|
|
is a multimodal minimization problem defined as follows:
|
||
|
|
|
||
|
|
.. math::
|
||
|
|
|
||
|
|
f_{\text{Plateau}}(x) = 30 + \sum_{i=1}^n \lfloor \lvert x_i
|
||
|
|
\rvert\rfloor
|
||
|
|
|
||
|
|
|
||
|
|
Here, :math:`n` represents the number of dimensions and
|
||
|
|
:math:`x_i \in [-5.12, 5.12]` for :math:`i = 1, ..., n`.
|
||
|
|
|
||
|
|
*Global optimum*: :math:`f(x) = 30` for :math:`x_i = 0` for
|
||
|
|
:math:`i = 1, ..., n`
|
||
|
|
|
||
|
|
.. [1] Gavana, A. Global Optimization Benchmarks and AMPGO retrieved 2015
|
||
|
|
"""
|
||
|
|
|
||
|
|
def __init__(self, dimensions=2):
|
||
|
|
Benchmark.__init__(self, dimensions)
|
||
|
|
|
||
|
|
self._bounds = list(zip([-5.12] * self.N, [5.12] * self.N))
|
||
|
|
|
||
|
|
self.global_optimum = [[0.0 for _ in range(self.N)]]
|
||
|
|
self.fglob = 30.0
|
||
|
|
self.change_dimensionality = True
|
||
|
|
|
||
|
|
def fun(self, x, *args):
|
||
|
|
self.nfev += 1
|
||
|
|
|
||
|
|
return 30.0 + sum(floor(abs(x)))
|
||
|
|
|
||
|
|
|
||
|
|
class Powell(Benchmark):
|
||
|
|
|
||
|
|
r"""
|
||
|
|
Powell objective function.
|
||
|
|
|
||
|
|
This class defines the Powell [1]_ global optimization problem. This
|
||
|
|
is a multimodal minimization problem defined as follows:
|
||
|
|
|
||
|
|
.. math::
|
||
|
|
|
||
|
|
f_{\text{Powell}}(x) = (x_3+10x_1)^2 + 5(x_2-x_4)^2 + (x_1-2x_2)^4
|
||
|
|
+ 10(x_3-x_4)^4
|
||
|
|
|
||
|
|
|
||
|
|
Here, :math:`n` represents the number of dimensions and
|
||
|
|
:math:`x_i \in [-4, 5]` for :math:`i = 1, ..., 4`.
|
||
|
|
|
||
|
|
*Global optimum*: :math:`f(x) = 0` for :math:`x_i = 0` for
|
||
|
|
:math:`i = 1, ..., 4`
|
||
|
|
|
||
|
|
..[1] Powell, M. An iterative method for finding stationary values of a
|
||
|
|
function of several variables Computer Journal, 1962, 5, 147-151
|
||
|
|
"""
|
||
|
|
|
||
|
|
def __init__(self, dimensions=4):
|
||
|
|
Benchmark.__init__(self, dimensions)
|
||
|
|
|
||
|
|
self._bounds = list(zip([-4.0] * self.N, [5.0] * self.N))
|
||
|
|
self.global_optimum = [[0, 0, 0, 0]]
|
||
|
|
self.fglob = 0
|
||
|
|
|
||
|
|
def fun(self, x, *args):
|
||
|
|
self.nfev += 1
|
||
|
|
|
||
|
|
return ((x[0] + 10 * x[1]) ** 2 + 5 * (x[2] - x[3]) ** 2
|
||
|
|
+ (x[1] - 2 * x[2]) ** 4 + 10 * (x[0] - x[3]) ** 4)
|
||
|
|
|
||
|
|
|
||
|
|
class PowerSum(Benchmark):
|
||
|
|
|
||
|
|
r"""
|
||
|
|
Power sum objective function.
|
||
|
|
|
||
|
|
This class defines the Power Sum global optimization problem. This
|
||
|
|
is a multimodal minimization problem defined as follows:
|
||
|
|
|
||
|
|
.. math::
|
||
|
|
|
||
|
|
f_{\text{PowerSum}}(x) = \sum_{k=1}^n\left[\left(\sum_{i=1}^n x_i^k
|
||
|
|
\right) - b_k \right]^2
|
||
|
|
|
||
|
|
Where, in this exercise, :math:`b = [8, 18, 44, 114]`
|
||
|
|
|
||
|
|
Here, :math:`x_i \in [0, 4]` for :math:`i = 1, ..., 4`.
|
||
|
|
|
||
|
|
*Global optimum*: :math:`f(x) = 0` for :math:`x = [1, 2, 2, 3]`
|
||
|
|
|
||
|
|
.. [1] Gavana, A. Global Optimization Benchmarks and AMPGO retrieved 2015
|
||
|
|
|
||
|
|
"""
|
||
|
|
|
||
|
|
def __init__(self, dimensions=4):
|
||
|
|
Benchmark.__init__(self, dimensions)
|
||
|
|
|
||
|
|
self._bounds = list(zip([0.0] * self.N,
|
||
|
|
[4.0] * self.N))
|
||
|
|
|
||
|
|
self.global_optimum = [[1.0, 2.0, 2.0, 3.0]]
|
||
|
|
self.fglob = 0.0
|
||
|
|
|
||
|
|
def fun(self, x, *args):
|
||
|
|
self.nfev += 1
|
||
|
|
|
||
|
|
b = [8.0, 18.0, 44.0, 114.0]
|
||
|
|
|
||
|
|
k = atleast_2d(arange(self.N) + 1).T
|
||
|
|
return sum((sum(x ** k, axis=1) - b) ** 2)
|
||
|
|
|
||
|
|
|
||
|
|
class Price01(Benchmark):
|
||
|
|
|
||
|
|
r"""
|
||
|
|
Price 1 objective function.
|
||
|
|
|
||
|
|
This class defines the Price 1 [1]_ global optimization problem. This
|
||
|
|
is a multimodal minimization problem defined as follows:
|
||
|
|
|
||
|
|
.. math::
|
||
|
|
|
||
|
|
f_{\text{Price01}}(x) = (\lvert x_1 \rvert - 5)^2
|
||
|
|
+ (\lvert x_2 \rvert - 5)^2
|
||
|
|
|
||
|
|
|
||
|
|
with :math:`x_i \in [-500, 500]` for :math:`i = 1, 2`.
|
||
|
|
|
||
|
|
*Global optimum*: :math:`f(x_i) = 0.0` for :math:`x = [5, 5]` or
|
||
|
|
:math:`x = [5, -5]` or :math:`x = [-5, 5]` or :math:`x = [-5, -5]`.
|
||
|
|
|
||
|
|
.. [1] Price, W. A controlled random search procedure for global
|
||
|
|
optimisation Computer Journal, 1977, 20, 367-370
|
||
|
|
"""
|
||
|
|
|
||
|
|
def __init__(self, dimensions=2):
|
||
|
|
Benchmark.__init__(self, dimensions)
|
||
|
|
|
||
|
|
self._bounds = list(zip([-500.0] * self.N,
|
||
|
|
[500.0] * self.N))
|
||
|
|
self.custom_bounds = ([-10.0, 10.0], [-10.0, 10.0])
|
||
|
|
|
||
|
|
self.global_optimum = [[5.0, 5.0]]
|
||
|
|
self.fglob = 0.0
|
||
|
|
|
||
|
|
def fun(self, x, *args):
|
||
|
|
self.nfev += 1
|
||
|
|
|
||
|
|
return (abs(x[0]) - 5.0) ** 2.0 + (abs(x[1]) - 5.0) ** 2.0
|
||
|
|
|
||
|
|
|
||
|
|
class Price02(Benchmark):
|
||
|
|
|
||
|
|
r"""
|
||
|
|
Price 2 objective function.
|
||
|
|
|
||
|
|
This class defines the Price 2 [1]_ global optimization problem. This is a
|
||
|
|
multimodal minimization problem defined as follows:
|
||
|
|
|
||
|
|
.. math::
|
||
|
|
|
||
|
|
f_{\text{Price02}}(x) = 1 + \sin^2(x_1) + \sin^2(x_2)
|
||
|
|
- 0.1e^{(-x_1^2 - x_2^2)}
|
||
|
|
|
||
|
|
|
||
|
|
with :math:`x_i \in [-10, 10]` for :math:`i = 1, 2`.
|
||
|
|
|
||
|
|
*Global optimum*: :math:`f(x) = 0.9` for :math:`x_i = [0, 0]`
|
||
|
|
|
||
|
|
.. [1] Price, W. A controlled random search procedure for global
|
||
|
|
optimisation Computer Journal, 1977, 20, 367-370
|
||
|
|
"""
|
||
|
|
|
||
|
|
def __init__(self, dimensions=2):
|
||
|
|
Benchmark.__init__(self, dimensions)
|
||
|
|
|
||
|
|
self._bounds = list(zip([-10.0] * self.N, [10.0] * self.N))
|
||
|
|
|
||
|
|
self.global_optimum = [[0.0, 0.0]]
|
||
|
|
self.fglob = 0.9
|
||
|
|
|
||
|
|
def fun(self, x, *args):
|
||
|
|
self.nfev += 1
|
||
|
|
|
||
|
|
return 1.0 + sum(sin(x) ** 2) - 0.1 * exp(-x[0] ** 2.0 - x[1] ** 2.0)
|
||
|
|
|
||
|
|
|
||
|
|
class Price03(Benchmark):
|
||
|
|
|
||
|
|
r"""
|
||
|
|
Price 3 objective function.
|
||
|
|
|
||
|
|
This class defines the Price 3 [1]_ global optimization problem. This is a
|
||
|
|
multimodal minimization problem defined as follows:
|
||
|
|
|
||
|
|
.. math::
|
||
|
|
|
||
|
|
f_{\text{Price03}}(x) = 100(x_2 - x_1^2)^2 + \left[6.4(x_2 - 0.5)^2
|
||
|
|
- x_1 - 0.6 \right]^2
|
||
|
|
|
||
|
|
with :math:`x_i \in [-50, 50]` for :math:`i = 1, 2`.
|
||
|
|
|
||
|
|
*Global optimum*: :math:`f(x) = 0` for :math:`x = [-5, -5]`,
|
||
|
|
:math:`x = [-5, 5]`, :math:`x = [5, -5]`, :math:`x = [5, 5]`.
|
||
|
|
|
||
|
|
.. [1] Price, W. A controlled random search procedure for global
|
||
|
|
optimisation Computer Journal, 1977, 20, 367-370
|
||
|
|
|
||
|
|
TODO Jamil #96 has an erroneous factor of 6 in front of the square brackets
|
||
|
|
"""
|
||
|
|
|
||
|
|
def __init__(self, dimensions=2):
|
||
|
|
Benchmark.__init__(self, dimensions)
|
||
|
|
|
||
|
|
self._bounds = list(zip([-5.0] * self.N, [5.0] * self.N))
|
||
|
|
self.custom_bounds = ([0, 2], [0, 2])
|
||
|
|
|
||
|
|
self.global_optimum = [[1.0, 1.0]]
|
||
|
|
self.fglob = 0.0
|
||
|
|
|
||
|
|
def fun(self, x, *args):
|
||
|
|
self.nfev += 1
|
||
|
|
|
||
|
|
return (100 * (x[1] - x[0] ** 2) ** 2
|
||
|
|
+ (6.4 * (x[1] - 0.5) ** 2 - x[0] - 0.6) ** 2)
|
||
|
|
|
||
|
|
|
||
|
|
class Price04(Benchmark):
|
||
|
|
|
||
|
|
r"""
|
||
|
|
Price 4 objective function.
|
||
|
|
|
||
|
|
This class defines the Price 4 [1]_ global optimization problem. This
|
||
|
|
is a multimodal minimization problem defined as follows:
|
||
|
|
|
||
|
|
.. math::
|
||
|
|
|
||
|
|
f_{\text{Price04}}(x) = (2 x_1^3 x_2 - x_2^3)^2
|
||
|
|
+ (6 x_1 - x_2^2 + x_2)^2
|
||
|
|
|
||
|
|
with :math:`x_i \in [-50, 50]` for :math:`i = 1, 2`.
|
||
|
|
|
||
|
|
*Global optimum*: :math:`f(x) = 0` for :math:`x = [0, 0]`,
|
||
|
|
:math:`x = [2, 4]` and :math:`x = [1.464, -2.506]`
|
||
|
|
|
||
|
|
.. [1] Price, W. A controlled random search procedure for global
|
||
|
|
optimisation Computer Journal, 1977, 20, 367-370
|
||
|
|
"""
|
||
|
|
|
||
|
|
def __init__(self, dimensions=2):
|
||
|
|
Benchmark.__init__(self, dimensions)
|
||
|
|
|
||
|
|
self._bounds = list(zip([-50.0] * self.N, [50.0] * self.N))
|
||
|
|
self.custom_bounds = ([0, 2], [0, 2])
|
||
|
|
|
||
|
|
self.global_optimum = [[2.0, 4.0]]
|
||
|
|
self.fglob = 0.0
|
||
|
|
|
||
|
|
def fun(self, x, *args):
|
||
|
|
self.nfev += 1
|
||
|
|
|
||
|
|
return ((2.0 * x[1] * x[0] ** 3.0 - x[1] ** 3.0) ** 2.0
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||
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+ (6.0 * x[0] - x[1] ** 2.0 + x[1]) ** 2.0)
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