# encoding=utf8
"""Sphere problems."""
import numpy as np
from niapy.problems.problem import Problem
__all__ = ['Sphere', 'Sphere2', 'Sphere3']
[docs]class Sphere(Problem):
r"""Implementation of Sphere functions.
Date: 2018
Authors: Iztok Fister Jr.
License: MIT
Function: **Sphere function**
:math:`f(\mathbf{x}) = \sum_{i=1}^D x_i^2`
**Input domain:**
The function can be defined on any input domain but it is usually
evaluated on the hypercube :math:`x_i ∈ [0, 10]`, for all :math:`i = 1, 2,..., D`.
**Global minimum:** :math:`f(x^*) = 0`, at :math:`x^* = (0,...,0)`
LaTeX formats:
Inline:
$f(\mathbf{x}) = \sum_{i=1}^D x_i^2$
Equation:
\begin{equation}f(\mathbf{x}) = \sum_{i=1}^D x_i^2 \end{equation}
Domain:
$0 \leq x_i \leq 10$
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=-5.12, upper=5.12, *args, **kwargs):
r"""Initialize Sphere 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'''$f(\mathbf{x}) = \sum_{i=1}^D x_i^2$'''
def _evaluate(self, x):
return np.sum(x ** 2)
[docs]class Sphere2(Problem):
r"""Implementation of Sphere with different powers function.
Date: 2018
Authors: Klemen Berkovič
License: MIT
Function: **Sun of different powers function**
:math:`f(\textbf{x}) = \sum_{i = 1}^D \lvert x_i \rvert^{i + 1}`
**Input domain:**
The function can be defined on any input domain but it is usually
evaluated on the hypercube :math:`x_i ∈ [-1, 1]`, for all :math:`i = 1, 2,..., D`.
**Global minimum:** :math:`f(x^*) = 0`, at :math:`x^* = (0,...,0)`
LaTeX formats:
Inline:
$f(\textbf{x}) = \sum_{i = 1}^D \lvert x_i \rvert^{i + 1}$
Equation:
\begin{equation} f(\textbf{x}) = \sum_{i = 1}^D \lvert x_i \rvert^{i + 1} \end{equation}
Domain:
$-1 \leq x_i \leq 1$
Reference URL:
https://www.sfu.ca/~ssurjano/sumpow.html
"""
[docs] def __init__(self, dimension=4, lower=-1.0, upper=1.0, *args, **kwargs):
r"""Initialize Sphere2 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'''$f(\textbf{x}) = \sum_{i = 1}^D \lvert x_i \rvert^{i + 1}$'''
def _evaluate(self, x):
indices = np.arange(2, self.dimension + 2)
return np.sum(np.power(np.abs(x), indices))
[docs]class Sphere3(Problem):
r"""Implementation of rotated hyper-ellipsoid function.
Date: 2018
Authors: Klemen Berkovič
License: MIT
Function: **Sun of rotated hyper-ellipsoid function**
:math:`f(\textbf{x}) = \sum_{i = 1}^D \sum_{j = 1}^i x_j^2`
**Input domain:**
The function can be defined on any input domain but it is usually
evaluated on the hypercube :math:`x_i ∈ [-65.536, 65.536]`, for all :math:`i = 1, 2,..., D`.
**Global minimum:** :math:`f(x^*) = 0`, at :math:`x^* = (0,...,0)`
LaTeX formats:
Inline:
$f(\textbf{x}) = \sum_{i = 1}^D \sum_{j = 1}^i x_j^2$
Equation:
\begin{equation} f(\textbf{x}) = \sum_{i = 1}^D \sum_{j = 1}^i x_j^2 \end{equation}
Domain:
$-65.536 \leq x_i \leq 65.536$
Reference URL:
https://www.sfu.ca/~ssurjano/rothyp.html
"""
[docs] def __init__(self, dimension=4, lower=-65.536, upper=65.536, *args, **kwargs):
r"""Initialize Sphere3 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'''$f(\textbf{x}) = \sum_{i = 1}^D \sum_{j = 1}^i x_j^2$'''
def _evaluate(self, x):
x_matrix = np.tile(x, (self.dimension, 1))
val = np.sum(np.tril(x_matrix) ** 2.0, axis=0)
return np.sum(val)
