# encoding=utf8
"""Implementations of Levy function."""
from NiaPy.benchmarks.benchmark import Benchmark
__all__ = ['Powell']
[docs]class Powell(Benchmark):
r"""Implementations of Powell functions.
Date:
2018
Author:
Klemen Berkovič
License:
MIT
Function:
**Levy Function**
:math:`f(\textbf{x}) = \sum_{i = 1}^{D / 4} \left( (x_{4 i - 3} + 10 x_{4 i - 2})^2 + 5 (x_{4 i - 1} - x_{4 i})^2 + (x_{4 i - 2} - 2 x_{4 i - 1})^4 + 10 (x_{4 i - 3} - x_{4 i})^4 \right)`
**Input domain:**
The function can be defined on any input domain but it is usually
evaluated on the hypercube :math:`x_i ∈ [-4, 5]`, for all :math:`i = 1, 2,..., D`.
**Global minimum:**
:math:`f(\textbf{x}^*) = 0` at :math:`\textbf{x}^* = (0, \cdots, 0)`
LaTeX formats:
Inline:
$f(\textbf{x}) = \sum_{i = 1}^{D / 4} \left( (x_{4 i - 3} + 10 x_{4 i - 2})^2 + 5 (x_{4 i - 1} - x_{4 i})^2 + (x_{4 i - 2} - 2 x_{4 i - 1})^4 + 10 (x_{4 i - 3} - x_{4 i})^4 \right)$
Equation:
\begin{equation} f(\textbf{x}) = \sum_{i = 1}^{D / 4} \left( (x_{4 i - 3} + 10 x_{4 i - 2})^2 + 5 (x_{4 i - 1} - x_{4 i})^2 + (x_{4 i - 2} - 2 x_{4 i - 1})^4 + 10 (x_{4 i - 3} - x_{4 i})^4 \right) \end{equation}
Domain:
$-4 \leq x_i \leq 5$
Attributes:
Name (List[str]): Names of the algorithm.
See Also:
* :class:`NiaPy.benchmarks.Benchmark`
Reference:
https://www.sfu.ca/~ssurjano/levy.html
"""
Name = ['Powell', 'powell']
[docs] def __init__(self, Lower=-4.0, Upper=5.0, **kwargs):
r"""Initialize of Powell benchmark.
Args:
Lower (Optional[float]): Lower bound of problem.
Upper (Optional[float]): Upper bound of problem.
kwargs (Dict[str, Any]): Additional arguments.
See Also:
:func:`NiaPy.benchmarks.Benchmark.__init__`
"""
Benchmark.__init__(self, Lower, Upper)
[docs] @staticmethod
def latex_code():
r"""Return the latex code of the problem.
Returns:
str: Latex code
"""
return r'''$f(\textbf{x}) = \sum_{i = 1}^{D / 4} \left( (x_{4 i - 3} + 10 x_{4 i - 2})^2 + 5 (x_{4 i - 1} - x_{4 i})^2 + (x_{4 i - 2} - 2 x_{4 i - 1})^4 + 10 (x_{4 i - 3} - x_{4 i})^4 \right)$'''
[docs] def function(self):
r"""Return benchmark evaluation function.
Returns:
Callable[[int, Union[int, float, list, numpy.ndarray], Dict[str, Any]], float]: Fitness function
"""
def f(D, X, **kwargs):
r"""Fitness function.
Args:
D (int): Dimensionality of the problem
X (Union[int, float, list, numpy.ndarray]): Solution to check.
kwargs (Dict[str, Any]): Additional arguments.
Returns:
float: Fitness value for the solution.
"""
v = 0.0
for i in range(1, (D // 4) + 1): v += (X[4 * i - 4] + 10 * X[4 * i - 3]) ** 2 + 5 * (X[4 * i - 2] - X[4 * i - 1]) ** 2 + (X[4 * i - 3] - 2 * X[4 * i - 2]) ** 4 + 10 * (X[4 * i - 4] - X[4 * i - 1]) ** 4
return v
return f