# encoding=utf8
"""Implementations of Weierstrass functions."""
from math import pi, cos
from NiaPy.benchmarks.benchmark import Benchmark
__all__ = ['Weierstrass']
[docs]class Weierstrass(Benchmark):
r"""Implementations of Weierstrass functions.
Date:
2018
Author:
Klemen Berkovič
License:
MIT
Function:
**Weierstass 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^* = (420.968746,...,420.968746)`
LaTeX formats:
Inline:
$$f(\textbf{x}) = \sum_{i=1}^D \left( \sum_{k=0}^{k_{max}} a^k \cos\left( 2 \pi b^k ( x_i + 0.5) \right) \right) - D \sum_{k=0}^{k_{max}} a^k \cos \left( 2 \pi b^k \cdot 0.5 \right)
Equation:
\begin{equation} f(\textbf{x}) = \sum_{i=1}^D \left( \sum_{k=0}^{k_{max}} a^k \cos\left( 2 \pi b^k ( x_i + 0.5) \right) \right) - D \sum_{k=0}^{k_{max}} a^k \cos \left( 2 \pi b^k \cdot 0.5 \right) \end{equation}
Domain:
$-100 \leq x_i \leq 100$
Reference:
http://www5.zzu.edu.cn/__local/A/69/BC/D3B5DFE94CD2574B38AD7CD1D12_C802DAFE_BC0C0.pdf
Attributes:
Name (List[str]): Names of the benchmark.
a (float): Benchmark parameter.
b (float): Benchmark parameter.
k_max (float): Benchmark parameter.
See Also:
* :class:`NiaPy.benchmarks.Benchmark`
"""
Name = ['Weierstrass', 'weierstrass']
a = 0.5
b = 3
k_max = 20
[docs] def __init__(self, Lower=-100.0, Upper=100.0, a=0.5, b=3, k_max=20, **kwargs):
r"""Initialize of Bent Cigar benchmark.
Args:
Lower (Optional[float]): Lower bound of problem.
Upper (Optional[float]): Upper bound of problem.
a (Optional[float]): Benchmark parameter.
b (Optional[float]): Benchmark parameter.
k_max (Optional[float]): Benchmark parameter.
kwargs (dict): Additional arguments.
See Also:
:func:`NiaPy.benchmarks.Benchmark.__init__`
"""
Benchmark.__init__(self, Lower, Upper)
self.a, self.b, self.k_max = a, b, k_max
[docs] @staticmethod
def latex_code():
r"""Return the latex code of the problem.
Returns:
str: Latex code
"""
return r'''$$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)'''
[docs] def function(self):
r"""Return benchmark evaluation function.
Returns:
Callable[[int, Union[int, float, list, numpy.ndarray], dict], float]: Fitness function
"""
self_a, self_b, self_k_max = self.a, self.b, self.k_max
def f(D, sol, a=None, b=None, k_max=None, **kwargs):
r"""Fitness function.
Args:
D (int): Dimensionality of the problem
sol (Union[int, float, list, numpy.ndarray]): Solution to check.
a (Optional[float]): Benchmark parameter.
b (Optional[float]): Benchmark parameter.
k_max (Optional[float]): Benchmark parameter.
kwargs (dict): Additional arguments.
Returns:
float: Fitness value for the solution.
"""
a = a if a is not None else self_a
b = a if b is not None else self_b
k_max = k_max if k_max is not None else self_k_max
val1 = 0.0
for i in range(D):
val = 0.0
for k in range(k_max): val += a ** k * cos(2 * pi * b ** k * (sol[i] + 0.5))
val1 += val
val2 = 0.0
for k in range(k_max): val2 += a ** k * cos(2 * pi * b ** k * 0.5)
return val1 - D * val2
return f