Source code for gemseo.problems.multiobjective_optimization.poloni
# Copyright 2021 IRT Saint Exupéry, https://www.irt-saintexupery.com
#
# 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
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
# 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 - initial API and implementation and/or initial documentation
# :author: Vincent Drouet
# OTHER AUTHORS - MACROSCOPIC CHANGES
# :author: François Gallard - minor improvements for integration
r"""Poloni's bi-objective optimization problem.
See :cite:`POLONI2000403`.
.. math::
\begin{aligned}
&a1 = 0.5 * sin(1) - 2 * cos(1) + sin(2) - 1.5 * cos(2)\\
&a2 = 1.5 * sin(1) - cos(1) + 2 * sin(2) - 0.5 * cos(2)\\
&b1(x, y) = 0.5 * sin(x) - 2 * cos(x) + sin(y) - 1.5 * cos(y)\\
&b2(x, y) = 1.5 * sin(x) - cos(x) + 2 * sin(y) - 0.5 * cos(y)\\
\text{minimize the objective function}\\
& f_1(x, y) = 1 + (a1 - b1(x,y)^2 + (a2 - b2(x,y))^2 \\
& f_2(x, y) = (x + 3)^2 + (y + 1)^2 \\
\text{with respect to the design variables}\\
&x \\
\text{subject to the bound constraints}\\
& -\pi \leq x \leq \pi\\
& -\pi \leq y \leq \pi
\end{aligned}
"""
from __future__ import annotations
from math import cos
from math import pi
from math import sin
from typing import TYPE_CHECKING
from numpy import array
from gemseo import create_design_space
from gemseo.algos.opt_problem import OptimizationProblem
from gemseo.core.mdofunctions.mdo_function import MDOFunction
if TYPE_CHECKING:
from gemseo.typing import RealArray
[docs]
class Poloni(OptimizationProblem):
"""Poloni multi-objective, bound constrained optimization problem."""
def __init__(self) -> None: # noqa: D205 D212 D107
design_space = create_design_space()
design_space.add_variable("x", l_b=-pi, u_b=pi, value=0)
design_space.add_variable("y", l_b=-pi, u_b=pi, value=0)
super().__init__(design_space)
self.objective = MDOFunction(
self._compute_output, self.__class__.__name__, jac=self._compute_jacobian
)
@staticmethod
def _compute_output(x: RealArray) -> RealArray:
"""Compute the output of the function.
Args:
x: The values to compute the output of the function.
Returns:
The output of the function.
"""
x, y = x
a1 = 0.5 * sin(1) - 2 * cos(1) + sin(2) - 1.5 * cos(2)
a2 = 1.5 * sin(1) - cos(1) + 2 * sin(2) - 0.5 * cos(2)
b1 = 0.5 * sin(x) - 2 * cos(x) + sin(y) - 1.5 * cos(y)
b2 = 1.5 * sin(x) - cos(x) + 2 * sin(y) - 0.5 * cos(y)
f2 = 1 + (a1 - b1) ** 2 + (a2 - b2) ** 2
f1 = (x + 3) ** 2 + (y + 1) ** 2
return array([f1, f2])
@staticmethod
def _compute_jacobian(x: RealArray) -> RealArray:
"""Compute the Jacobian of the function.
Args:
x: The values to compute the Jacobian of the function.
Returns:
The Jacobian value of the function.
"""
x, y = x
a1 = 0.5 * sin(1) - 2 * cos(1) + sin(2) - 1.5 * cos(2)
a2 = 1.5 * sin(1) - cos(1) + 2 * sin(2) - 0.5 * cos(2)
b1 = 0.5 * sin(x) - 2 * cos(x) + sin(y) - 1.5 * cos(y)
b2 = 1.5 * sin(x) - cos(x) + 2 * sin(y) - 0.5 * cos(y)
amb1 = a1 - b1
amb2 = a2 - b2
df2_dx = -amb1 * (cos(x) + 4 * sin(x)) - amb2 * (3 * cos(x) + 2 * sin(x))
df2_dy = -amb1 * (2 * cos(y) + sin(x)) - amb2 * (4 * cos(y) + sin(y))
df1_dx = 2 * (x + 3)
df1_dy = 2 * (y + 1)
return array([[df1_dx, df1_dy], [df2_dx, df2_dy]])
