Source code for gemseo.problems.analytical.rastrigin

# -*- coding: utf-8 -*-
# Copyright 2021 IRT Saint Exupéry,
# This program is free software; you can redistribute it and/or
# modify it under the terms of the GNU Lesser General Public
# License version 3 as published by the Free Software Foundation.
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# but WITHOUT ANY WARRANTY; without even the implied warranty of
# Lesser General Public License for more details.
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# Inc., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301, USA.

# Contributors:
#    INITIAL AUTHORS - API and implementation and/or documentation
#        :author: Damien Guenot
#        :author: Francois Gallard
The Rastrigin analytic problem
from __future__ import division, unicode_literals

import logging
from cmath import cos, pi, sin

from numpy import array, ones, zeros

from gemseo.algos.design_space import DesignSpace
from gemseo.algos.opt_problem import OptimizationProblem
from gemseo.core.mdofunctions.mdo_function import MDOFunction

LOGGER = logging.getLogger(__name__)

[docs]class Rastrigin(OptimizationProblem): r"""**Rastrigin** :class:`.OptimizationProblem` uses the Rastrigin objective function with the :class:`.DesignSpace` :math:`[-0.1,0.1]^2` From the Rastrigin function is a non-convex function used as a performance test problem for optimization algorithms. It is a typical example of non-linear multimodal function. It was first proposed by [Rastrigin] as a 2-dimensional function and has been generalized by [MuhlenbeinEtAl]. Finding the minimum of this function is a fairly difficult problem due to its large search space and its large number of local minima. It has a global minimum at :math:`x=0` where :math:`f(x)=0`. It can be extended to :math:`n>2` dimensions: .. math:: f(x) = 10n + \sum_{i=1}^n [x_i^2 - 10\cos(2\pi x_i)] [Rastrigin] Rastrigin, L. A. "Systems of extremal control." Mir, Moscow (1974). [MuhlenbeinEtAl] H. Mühlenbein, D. Schomisch and J. Born. "The Parallel Genetic Algorithm as Function Optimizer ". Parallel Computing, 17, pages 619–632, 1991. """ def __init__(self): """ The constructor initializes the Rastrigin :class:`.OptimizationProblem` by defining the :class:`.DesignSpace` and the objective function. """ design_space = DesignSpace() design_space.add_variable("x", 2, l_b=-0.1, u_b=0.1) design_space.set_current_x(0.01 * ones(2)) super(Rastrigin, self).__init__(design_space) expr = "20 + sum(x[i]**2 - 10*cos(2pi*x[i]))" self.objective = MDOFunction( self.rastrigin, name="Rastrigin", f_type="obj", jac=self.rastrigin_jac, expr=expr, args=["x"], )
[docs] @staticmethod def rastrigin(x_dv): """This function computes the order n=2 Rastrigin function. :param x_dv: design variable vector of size 2 :returns: result of Rastrigin function evaluation """ a_c = 10.0 func = ( a_c * 2.0 + (x_dv[0] ** 2 - a_c * cos(2 * pi * x_dv[0])) + (x_dv[1] ** 2 - a_c * cos(2 * pi * x_dv[1])) ) return func.real
[docs] @staticmethod def get_solution(): """Return theoretical optimal value of Rastrigin function. :returns: design variables values of optimized values, function value at optimum :rtype: numpy array """ x_opt = zeros(2) f_opt = 0.0 return x_opt, f_opt
[docs] @staticmethod def rastrigin_jac(x_dv): """This function computes the analytical gradient of 2nd order Rastrigin function. :param x_dv: design variable vector :type x_dv: numpy array :returns: analytical gradient vector of Rastrigin function :rtype: numpy array """ a_c = 10.0 analytic_grad = array( [ 2 * x_dv[0] + 2 * pi * a_c * sin(2 * pi * x_dv[0]), 2 * x_dv[1] + 2 * pi * a_c * sin(2 * pi * x_dv[1]), ] ) return analytic_grad.real