.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "examples/multi_objective/plot_mnbi_fonseca.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_examples_multi_objective_plot_mnbi_fonseca.py>`
        to download the full example code

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_examples_multi_objective_plot_mnbi_fonseca.py:


Multi-objective Fonseca-Fleming example with the mNBI algorithm
===============================================================

In this example, the modified Normal Boundary Intersection algorithm (mNBI) is used to
solve the :class:`.FonsecaFleming` optimization problem :cite:`fonseca1995overview`:

.. math::

   \begin{aligned}
   \text{minimize the objective function }
   & f_1(x) = 1 - exp(-\sum_{i=1}^{d}((x_i - 1 / sqrt(d)) ^ 2)) \\
   & f_2(x) = 1 + exp(-\sum_{i=1}^{d}((x_i + 1 / sqrt(d)) ^ 2)) \\
   \text{with respect to the design variables }&x\\
   \text{subject to the bound constraint}
   & x\in[-4,4]^d
   \end{aligned}

We also show how the Pareto front can be refined.

.. GENERATED FROM PYTHON SOURCE LINES 40-51

.. code-block:: Python


    from __future__ import annotations

    from gemseo import configure_logger
    from gemseo import execute_algo
    from gemseo import execute_post
    from gemseo.problems.optimization.fonseca_fleming import FonsecaFleming

    configure_logger()




.. rst-class:: sphx-glr-script-out

.. code-block:: pytb

    Traceback (most recent call last):
      File "/home/docs/checkouts/readthedocs.org/user_builds/gemseo/checkouts/develop/doc_src/_examples/multi_objective/plot_mnbi_fonseca.py", line 46, in <module>
        from gemseo.problems.optimization.fonseca_fleming import FonsecaFleming
    ModuleNotFoundError: No module named 'gemseo.problems.optimization.fonseca_fleming'




.. GENERATED FROM PYTHON SOURCE LINES 52-58

Solve the Fonseca-Fleming optimization problem
----------------------------------------------
The 3 sub-optimization problems of mNBI are solved with SLSQP,
a gradient-based optimization algorithm from the NLOPT library,
with a maximum of 100 iterations.
The analytic gradients are provided.

.. GENERATED FROM PYTHON SOURCE LINES 58-67

.. code-block:: Python

    opt_problem = FonsecaFleming()
    result = execute_algo(
        opt_problem,
        "MNBI",
        max_iter=1000,
        sub_optim_max_iter=100,
        n_sub_optim=3,
        sub_optim_algo="NLOPT_SLSQP",
    )

.. GENERATED FROM PYTHON SOURCE LINES 68-74

Display the Pareto front
^^^^^^^^^^^^^^^^^^^^^^^^
|g| detects the Pareto optimal points and the dominated ones.
The Fonseca-Fleming problem is interesting because
its Pareto front is not convex.
The mNBI algorithm successfully computes it.

.. GENERATED FROM PYTHON SOURCE LINES 74-77

.. code-block:: Python


    execute_post(opt_problem, "ParetoFront", save=False, show=True)


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Solve the Fonseca-Fleming optimization problem more finely
----------------------------------------------------------
The Pareto front is then refined with 10 sub-optimizations instead of 3.

.. GENERATED FROM PYTHON SOURCE LINES 81-90

.. code-block:: Python

    opt_problem = FonsecaFleming()
    result = execute_algo(
        opt_problem,
        "MNBI",
        max_iter=1000,
        sub_optim_max_iter=100,
        n_sub_optim=10,
        sub_optim_algo="NLOPT_SLSQP",
    )

.. GENERATED FROM PYTHON SOURCE LINES 91-94

Display the Pareto front
^^^^^^^^^^^^^^^^^^^^^^^^
We can clearly see the effect of the refinement.

.. GENERATED FROM PYTHON SOURCE LINES 94-96

.. code-block:: Python


    execute_post(opt_problem, "ParetoFront", save=False, show=True)


.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.002 seconds)


.. _sphx_glr_download_examples_multi_objective_plot_mnbi_fonseca.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_mnbi_fonseca.ipynb <plot_mnbi_fonseca.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_mnbi_fonseca.py <plot_mnbi_fonseca.py>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_