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

.. only:: html

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

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

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

.. _sphx_glr_examples_post_process_algorithms_plot_quad_approx.py:


Quadratic approximations
========================

In this example, we illustrate the use of the :class:`.QuadApprox` plot
on the Sobieski's SSBJ problem.

The :class:`.QuadApprox` post-processing performs a quadratic approximation of a given
function from an optimization history and provide two plots.

The first plot shows an approximation of the Hessian matrix
:math:`\dfrac{\partial^2 f}{\partial x_i \partial x_j}` based on the *Symmetric Rank 1*
method (SR1) :cite:`Nocedal2006`. The color map uses a symmetric logarithmic (symlog)
scale. This plots the cross influence of the design variables on the objective function
or constraints. For instance, on the last figure, the maximal second-order sensitivity
is :math:`\dfrac{\partial^2 -y_4}{\partial^2 x_0} = 2.10^5`, which means that the
:math:`x_0` is the most influential variable. Then, the cross derivative
:math:`\dfrac{\partial^2 -y_4}{\partial x_0 \partial x_2} = 5.10^4` is positive and
relatively high compared to the previous one but the combined effects of :math:`x_0` and
:math:`x_2` are non-negligible in comparison.

The second plot represents the quadratic approximation of the objective around the
optimal solution : :math:`a_{i}(t)=0.5 (t-x^*_i)^2
\dfrac{\partial^2 f}{\partial x_i^2} + (t-x^*_i) \dfrac{\partial
f}{\partial x_i} + f(x^*)`, where :math:`x^*` is the optimal solution.
This approximation highlights the sensitivity of the :term:`objective function` with
respect to the :term:`design variables`: we notice that the design
variables :math:`x\_1, x\_5, x\_6` have little influence , whereas
:math:`x\_0, x\_2, x\_9` have a huge influence on the objective. This trend is also
noted in the diagonal terms of the :term:`Hessian` matrix
:math:`\dfrac{\partial^2 f}{\partial x_i^2}`.

.. GENERATED FROM PYTHON SOURCE LINES 53-63



.. rst-class:: sphx-glr-horizontal


    *

      .. image-sg:: /examples/post_process/algorithms/images/sphx_glr_plot_quad_approx_001.png
         :alt: Hessian matrix SR1 approximation of -y_4
         :srcset: /examples/post_process/algorithms/images/sphx_glr_plot_quad_approx_001.png
         :class: sphx-glr-multi-img

    *

      .. image-sg:: /examples/post_process/algorithms/images/sphx_glr_plot_quad_approx_002.png
         :alt: plot quad approx
         :srcset: /examples/post_process/algorithms/images/sphx_glr_plot_quad_approx_002.png
         :class: sphx-glr-multi-img


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

 .. code-block:: none

        INFO - 16:25:50: Importing the optimization problem from the file sobieski_mdf_scenario.h5

    <gemseo.post.quad_approx.QuadApprox object at 0x72a4e114fef0>





|

.. code-block:: Python


    from __future__ import annotations

    from gemseo import execute_post
    from gemseo.settings.post import QuadApprox_Settings

    execute_post(
        "sobieski_mdf_scenario.h5",
        settings_model=QuadApprox_Settings(function="-y_4", save=False, show=True),
    )


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

   **Total running time of the script:** (0 minutes 0.511 seconds)


.. _sphx_glr_download_examples_post_process_algorithms_plot_quad_approx.py:

.. only:: html

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

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

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

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

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

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_quad_approx.zip <plot_quad_approx.zip>`


.. only:: html

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

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