# encoding=utf8
"""Implementations of Weierstrass functions."""
import numpy as np
from niapy.problems.problem import Problem
__all__ = ['Weierstrass']
[docs]
class Weierstrass(Problem):
r"""Implementations of Weierstrass functions.
Date: 2018
Author: Klemen Berkovič
License: MIT
Function:
**Weierstrass Function**
:math:`f(\textbf{x}) = \sum_{i=1}^D \left( \sum_{k=0}^{k_{max}} a^k \cos\left( 2 \pi b^k ( x_i + 0.5) \right) \right) - D \sum_{k=0}^{k_{max}} a^k \cos \left( 2 \pi b^k \cdot 0.5 \right)`
**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`.
Default value of a = 0.5, b = 3 and k_max = 20.
**Global minimum:** :math:`f(x^*) = 0`, at :math:`x_i^* = 0`
LaTeX formats:
Inline:
$$f(\textbf{x}) = \sum_{i=1}^D \left( \sum_{k=0}^{k_{max}} a^k \cos\left( 2 \pi b^k ( x_i + 0.5) \right) \right) - D \sum_{k=0}^{k_{max}} a^k \cos \left( 2 \pi b^k \cdot 0.5 \right)
Equation:
\begin{equation} f(\textbf{x}) = \sum_{i=1}^D \left( \sum_{k=0}^{k_{max}} a^k \cos\left( 2 \pi b^k ( x_i + 0.5) \right) \right) - D \sum_{k=0}^{k_{max}} a^k \cos \left( 2 \pi b^k \cdot 0.5 \right) \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, a=0.5, b=3, k_max=20, *args, **kwargs):
r"""Initialize Weierstrass 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.
a (Optional[float]): The a parameter.
b (Optional[float]): The b parameter.
k_max (Optional[int]): Number of elements of the series to compute.
See Also:
:func:`niapy.problems.Problem.__init__`
"""
super().__init__(dimension, lower, upper, *args, **kwargs)
self.a = a
self.b = b
self.k_max = k_max
[docs]
@staticmethod
def latex_code():
r"""Return the latex code of the problem.
Returns:
str: Latex code.
"""
return r'''$f(\textbf{x}) = \sum_{i=1}^D \left( \sum_{k=0}^{k_{max}} a^k \cos\left( 2 \pi b^k ( x_i + 0.5) \right) \right) - D \sum_{k=0}^{k_{max}} a^k \cos \left( 2 \pi b^k \cdot 0.5 \right)$'''
def _evaluate(self, x):
k = np.atleast_2d(np.arange(self.k_max + 1)).T
t1 = self.a ** k * np.cos(2 * np.pi * self.b ** k * (x + 0.5))
t2 = self.dimension * np.sum(self.a ** k.T * np.cos(np.pi * self.b ** k.T))
return np.sum(np.sum(t1, axis=0)) - t2