Source code for niapy.problems.levy

# encoding=utf8

"""Implementations of Levy function."""

import numpy as np
from niapy.problems.problem import Problem

__all__ = ['Levy']


class Levy(Problem):
    r"""Implementations of Levy functions.

    Date: 2018

    Author: Klemen Berkovič

    License: MIT

    Function:
    **Levy Function**

        :math:`f(\textbf{x}) = \sin^2 (\pi w_1) + \sum_{i = 1}^{D - 1} (w_i - 1)^2 \left( 1 + 10 \sin^2 (\pi w_i + 1) \right) + (w_d - 1)^2 (1 + \sin^2 (2 \pi w_d)) \\ w_i = 1 + \frac{x_i - 1}{4}`

        **Input domain:**
        The function can be defined on any input domain but it is usually
        evaluated on the hypercube :math:`x_i ∈ [-10, 10]`, for all :math:`i = 1, 2,..., D`.

        **Global minimum:**
        :math:`f(\textbf{x}^*) = 0` at :math:`\textbf{x}^* = (1, \cdots, 1)`

    LaTeX formats:
        Inline:
            $f(\textbf{x}) = \sin^2 (\pi w_1) + \sum_{i = 1}^{D - 1} (w_i - 1)^2 \left( 1 + 10 \sin^2 (\pi w_i + 1) \right) + (w_d - 1)^2 (1 + \sin^2 (2 \pi w_d)) \\ w_i = 1 + \frac{x_i - 1}{4}$

        Equation:
            \begin{equation} f(\textbf{x}) = \sin^2 (\pi w_1) + \sum_{i = 1}^{D - 1} (w_i - 1)^2 \left( 1 + 10 \sin^2 (\pi w_i + 1) \right) + (w_d - 1)^2 (1 + \sin^2 (2 \pi w_d)) \\ w_i = 1 + \frac{x_i - 1}{4} \end{equation}

        Domain:
            $-10 \leq x_i \leq 10$

    Reference:
        https://www.sfu.ca/~ssurjano/levy.html

    """

    def __init__(self, dimension=4, lower=-10.0, upper=10.0, *args, **kwargs):
        r"""Initialize Levy 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)

    @staticmethod
    def latex_code():
        r"""Return the latex code of the problem.

        Returns:
            str: Latex code.

        """
        return r'''$f(\textbf{x}) = \sin^2 (\pi w_1) + \sum_{i = 1}^{D - 1} (w_i - 1)^2 \left( 1 + 10 \sin^2 (\pi w_i + 1) \right) + (w_d - 1)^2 (1 + \sin^2 (2 \pi w_d)) \\ w_i = 1 + \frac{x_i - 1}{4}$'''

    def _evaluate(self, x):
        w = 1 + (x - 1) / 4
        term1 = np.sin(np.pi * w[0]) ** 2
        wi = w[:-1]
        term2 = np.sum((wi - 1) ** 2 * (1 + 10 * np.sin(np.pi * wi + 1)))
        term3 = (w[-1] - 1) ** 2 * (1 + np.sin(2 * np.pi * w[-1]) ** 2)
        return term1 + term2 + term3