# encoding=utf8
"""Implementations of Schwefel's functions."""
import numpy as np
from niapy.problems.problem import Problem
__all__ = ['Schwefel', 'Schwefel221', 'Schwefel222', 'ModifiedSchwefel']
[docs]class Schwefel(Problem):
r"""Implementation of Schwefel function.
Date: 2018
Author: Lucija Brezočnik
License: MIT
Function: **Schwefel function**
:math:`f(\textbf{x}) = 418.9829d - \sum_{i=1}^{D} x_i \sin(\sqrt{\lvert x_i \rvert})`
**Input domain:**
The function can be defined on any input domain but it is usually
evaluated on the hypercube :math:`x_i ∈ [-500, 500]`, for all :math:`i = 1, 2,..., D`.
**Global minimum:** :math:`f(x^*) = 0`, at :math:`x^* = (420.968746,...,420.968746)`
LaTeX formats:
Inline:
$f(\textbf{x}) = 418.9829d - \sum_{i=1}^{D} x_i \sin(\sqrt{\lvert x_i \rvert})$
Equation:
\begin{equation} f(\textbf{x}) = 418.9829d - \sum_{i=1}^{D} x_i
\sin(\sqrt{\lvert x_i \rvert}) \end{equation}
Domain:
$-500 \leq x_i \leq 500$
Reference:
https://www.sfu.ca/~ssurjano/schwef.html
"""
[docs] def __init__(self, dimension=4, lower=-500.0, upper=500.0, *args, **kwargs):
r"""Initialize Schwefel problem..
Args:
dimension (Optional[int]): Dimension of the problem.
lower (Optional[Union[float, Iterable[float]]]): Lower bounds of the problem.
upper (Optional[Union[float, Iterable[float]]]): Upper bounds of the problem.
See Also:
:func:`niapy.problems.Problem.__init__`
"""
super().__init__(dimension, lower, upper, *args, **kwargs)
[docs] @staticmethod
def latex_code():
r"""Return the latex code of the problem.
Returns:
str: Latex code.
"""
return r'''$f(\textbf{x}) = 418.9829d - \sum_{i=1}^{D} x_i \sin(\sqrt{\lvert x_i \rvert})$'''
def _evaluate(self, x):
return 418.9829 * self.dimension - np.sum(x * np.sin(np.sqrt(np.abs(x))))
[docs]class Schwefel221(Problem):
r"""Schwefel 2.21 function implementation.
Date: 2018
Author: Grega Vrbančič
Licence: MIT
Function: **Schwefel 2.21 function**
:math:`f(\mathbf{x})=\max_{i=1,...,D}|x_i|`
**Input domain:**
The function can be defined on any input domain but it is usually
evaluated on the hypercube :math:`x_i ∈ [-100, 100]`, for all :math:`i = 1, 2,..., D`.
**Global minimum:** :math:`f(x^*) = 0`, at :math:`x^* = (0,...,0)`
LaTeX formats:
Inline:
$f(\mathbf{x})=\max_{i=1,...,D} \lvert x_i \rvert$
Equation:
\begin{equation}f(\mathbf{x}) = \max_{i=1,...,D} \lvert x_i \rvert \end{equation}
Domain:
$-100 \leq x_i \leq 100$
Reference paper:
Jamil, M., and Yang, X. S. (2013).
A literature survey of benchmark functions for global optimisation problems.
International Journal of Mathematical Modelling and Numerical Optimisation,
4(2), 150-194.
"""
[docs] def __init__(self, dimension=4, lower=-100.0, upper=100.0, *args, **kwargs):
r"""Initialize Schwefel221 problem..
Args:
dimension (Optional[int]): Dimension of the problem.
lower (Optional[Union[float, Iterable[float]]]): Lower bounds of the problem.
upper (Optional[Union[float, Iterable[float]]]): Upper bounds of the problem.
See Also:
:func:`niapy.problems.Problem.__init__`
"""
super().__init__(dimension, lower, upper, *args, **kwargs)
[docs] @staticmethod
def latex_code():
r"""Return the latex code of the problem.
Returns:
str: Latex code.
"""
return r'''$f(\mathbf{x})=\max_{i=1,...,D} \lvert x_i \rvert$'''
def _evaluate(self, x):
return np.amax(np.abs(x))
[docs]class Schwefel222(Problem):
r"""Schwefel 2.22 function implementation.
Date: 2018
Author: Grega Vrbančič
Licence: MIT
Function: **Schwefel 2.22 function**
:math:`f(\mathbf{x})=\sum_{i=1}^{D} \lvert x_i \rvert +\prod_{i=1}^{D} \lvert x_i \rvert`
**Input domain:**
The function can be defined on any input domain but it is usually
evaluated on the hypercube :math:`x_i ∈ [-100, 100]`, for all :math:`i = 1, 2,..., D`.
**Global minimum:** :math:`f(x^*) = 0`, at :math:`x^* = (0,...,0)`
LaTeX formats:
Inline:
$f(\mathbf{x})=\sum_{i=1}^{D} \lvert x_i \rvert +\prod_{i=1}^{D} \lvert x_i \rvert$
Equation:
\begin{equation}f(\mathbf{x}) = \sum_{i=1}^{D} \lvert x_i \rvert + \prod_{i=1}^{D} \lvert x_i \rvert \end{equation}
Domain:
$-100 \leq x_i \leq 100$
Reference paper:
Jamil, M., and Yang, X. S. (2013).
A literature survey of benchmark functions for global optimisation problems.
International Journal of Mathematical Modelling and Numerical Optimisation,
4(2), 150-194.
"""
[docs] def __init__(self, dimension=4, lower=-100.0, upper=100.0, *args, **kwargs):
r"""Initialize Schwefel222 problem..
Args:
dimension (Optional[int]): Dimension of the problem.
lower (Optional[Union[float, Iterable[float]]]): Lower bounds of the problem.
upper (Optional[Union[float, Iterable[float]]]): Upper bounds of the problem.
See Also:
:func:`niapy.problems.Problem.__init__`
"""
super().__init__(dimension, lower, upper, *args, **kwargs)
[docs] @staticmethod
def latex_code():
r"""Return the latex code of the problem.
Returns:
str: Latex code.
"""
return r'''$f(\mathbf{x})=\sum_{i=1}^{D} \lvert x_i \rvert +\prod_{i=1}^{D} \lvert x_i \rvert$'''
def _evaluate(self, x):
return np.sum(np.abs(x)) + np.product(np.abs(x))
[docs]class ModifiedSchwefel(Problem):
r"""Implementations of Modified Schwefel functions.
Date: 2018
Author: Klemen Berkovič
License: MIT
Function:
**Modified Schwefel Function**
:math:`f(\textbf{x}) = 418.9829 \cdot D - \sum_{i=1}^D h(x_i) \\ h(x) = g(x + 420.9687462275036) \\ g(z) = \begin{cases} z \sin \left( \lvert z \rvert^{\frac{1}{2}} \right) &\quad \lvert z \rvert \leq 500 \\ \left( 500 - \mod (z, 500) \right) \sin \left( \sqrt{\lvert 500 - \mod (z, 500) \rvert} \right) - \frac{ \left( z - 500 \right)^2 }{ 10000 D } &\quad z > 500 \\ \left( \mod (\lvert z \rvert, 500) - 500 \right) \sin \left( \sqrt{\lvert \mod (\lvert z \rvert, 500) - 500 \rvert} \right) + \frac{ \left( z - 500 \right)^2 }{ 10000 D } &\quad z < -500\end{cases}`
**Input domain:**
The function can be defined on any input domain but it is usually
evaluated on the hypercube :math:`x_i ∈ [-100, 100]`, for all :math:`i = 1, 2,..., D`.
**Global minimum:** :math:`f(x^*) = 0`, at :math:`x^* = (420.968746,...,420.968746)`
LaTeX formats:
Inline:
$f(\textbf{x}) = 418.9829 \cdot D - \sum_{i=1}^D h(x_i) \\ h(x) = g(x + 420.9687462275036) \\ g(z) = \begin{cases} z \sin \left( \lvert z \rvert^{\frac{1}{2}} \right) &\quad \lvert z \rvert \leq 500 \\ \left( 500 - \mod (z, 500) \right) \sin \left( \sqrt{\lvert 500 - \mod (z, 500) \rvert} \right) - \frac{ \left( z - 500 \right)^2 }{ 10000 D } &\quad z > 500 \\ \left( \mod (\lvert z \rvert, 500) - 500 \right) \sin \left( \sqrt{\lvert \mod (\lvert z \rvert, 500) - 500 \rvert} \right) + \frac{ \left( z - 500 \right)^2 }{ 10000 D } &\quad z < -500\end{cases}$
Equation:
\begin{equation} f(\textbf{x}) = 418.9829 \cdot D - \sum_{i=1}^D h(x_i) \\ h(x) = g(x + 420.9687462275036) \\ g(z) = \begin{cases} z \sin \left( \lvert z \rvert^{\frac{1}{2}} \right) &\quad \lvert z \rvert \leq 500 \\ \left( 500 - \mod (z, 500) \right) \sin \left( \sqrt{\lvert 500 - \mod (z, 500) \rvert} \right) - \frac{ \left( z - 500 \right)^2 }{ 10000 D } &\quad z > 500 \\ \left( \mod (\lvert z \rvert, 500) - 500 \right) \sin \left( \sqrt{\lvert \mod (\lvert z \rvert, 500) - 500 \rvert} \right) + \frac{ \left( z - 500 \right)^2 }{ 10000 D } &\quad z < -500\end{cases} \end{equation}
Domain:
$-100 \leq x_i \leq 100$
Reference:
http://www5.zzu.edu.cn/__local/A/69/BC/D3B5DFE94CD2574B38AD7CD1D12_C802DAFE_BC0C0.pdf
"""
[docs] def __init__(self, dimension=4, lower=-100.0, upper=100.0, *args, **kwargs):
r"""Initialize Modified Schwefel problem..
Args:
dimension (Optional[int]): Dimension of the problem.
lower (Optional[Union[float, Iterable[float]]]): Lower bounds of the problem.
upper (Optional[Union[float, Iterable[float]]]): Upper bounds of the problem.
See Also:
:func:`niapy.problems.Problem.__init__`
"""
super().__init__(dimension, lower, upper, *args, **kwargs)
[docs] @staticmethod
def latex_code():
r"""Return the latex code of the problem.
Returns:
str: Latex code.
"""
return r'''$f(\textbf{x}) = 418.9829 \cdot D - \sum_{i=1}^D h(x_i) \\ h(x) = g(x + 420.9687462275036) \\ g(z) = \begin{cases} z \sin \left( \lvert z \rvert^{\frac{1}{2}} \right) &\quad \lvert z \rvert \leq 500 \\ \left( 500 - \mod (z, 500) \right) \sin \left( \sqrt{\lvert 500 - \mod (z, 500) \rvert} \right) - \frac{ \left( z - 500 \right)^2 }{ 10000 D } &\quad z > 500 \\ \left( \mod (\lvert z \rvert, 500) - 500 \right) \sin \left( \sqrt{\lvert \mod (\lvert z \rvert, 500) - 500 \rvert} \right) + \frac{ \left( z - 500 \right)^2 }{ 10000 D } &\quad z < -500\end{cases}$'''
def _evaluate(self, x):
xx = x + 420.9687462275036
conditions = [x > 500.0, x < -500.0]
xx_mod = np.fmod(xx, 500.0)
choices = [(500.0 - xx_mod) * np.sin(np.sqrt(np.abs(500.0 - xx_mod))) - (xx - 500.0) ** 2 / (10000 * self.dimension),
(xx_mod - 500.0) * np.sin(np.sqrt(np.abs(xx_mod - 500.0))) + (xx - 500.0) ** 2 / (10000 * self.dimension)]
default = xx * np.sin(np.sqrt(np.abs(xx)))
val = np.sum(np.select(conditions, choices, default=default))
return 418.9829 * self.dimension - val
