Source code for gemseo.problems.analytical.rastrigin

# -*- coding: utf-8 -*-
# Copyright 2021 IRT Saint Exupéry, https://www.irt-saintexupery.com
#
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# Lesser General Public License for more details.
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# You should have received a copy of the GNU Lesser General Public License
# along with this program; if not, write to the Free Software Foundation,
# 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
#    OTHER AUTHORS   - MACROSCOPIC CHANGES
"""
The Rastrigin analytic problem
******************************
"""
from __future__ import absolute_import, division, print_function, unicode_literals

from cmath import cos, pi, sin

from future import standard_library
from numpy import array, ones, zeros

from gemseo.algos.design_space import DesignSpace
from gemseo.algos.opt_problem import OptimizationProblem
from gemseo.core.function import MDOFunction

standard_library.install_aliases()

from gemseo import LOGGER

[docs]class Rastrigin(OptimizationProblem):

r"""**Rastrigin** :class:.OptimizationProblem
uses the Rastrigin objective function
with the :class:.DesignSpace :math:[-0.1,0.1]^2

From http://en.wikipedia.org/wiki/Rastrigin_function:

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.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