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.. only:: html

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        :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.

.. GENERATED FROM PYTHON SOURCE LINES 28-35

.. code-block:: default

    from __future__ import annotations

    from gemseo import configure_logger
    from gemseo import create_discipline
    from gemseo import create_scenario
    from gemseo.problems.sobieski.core.problem import SobieskiProblem








.. GENERATED FROM PYTHON SOURCE LINES 36-40

Import
------
The first step is to import some high-level functions
and a method to get the design space.

.. GENERATED FROM PYTHON SOURCE LINES 40-43

.. code-block:: default


    configure_logger()





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

 .. code-block:: none


    <RootLogger root (INFO)>



.. GENERATED FROM PYTHON SOURCE LINES 44-51

Description
-----------

The :class:`.QuadApprox` post-processing
performs a quadratic approximation of a given function
from an optimization history
and plot the results as cuts of the approximation.

.. GENERATED FROM PYTHON SOURCE LINES 53-57

Create disciplines
------------------
Then, we instantiate the disciplines of the Sobieski's SSBJ problem:
Propulsion, Aerodynamics, Structure and Mission

.. GENERATED FROM PYTHON SOURCE LINES 57-66

.. code-block:: default

    disciplines = create_discipline(
        [
            "SobieskiPropulsion",
            "SobieskiAerodynamics",
            "SobieskiStructure",
            "SobieskiMission",
        ]
    )








.. GENERATED FROM PYTHON SOURCE LINES 67-70

Create design space
-------------------
We also read the design space from the :class:`.SobieskiProblem`.

.. GENERATED FROM PYTHON SOURCE LINES 70-72

.. code-block:: default

    design_space = SobieskiProblem().design_space








.. GENERATED FROM PYTHON SOURCE LINES 73-80

Create and execute scenario
---------------------------
The next step is to build an MDO scenario in order to maximize the range,
encoded 'y_4', with respect to the design parameters, while satisfying the
inequality constraints 'g_1', 'g_2' and 'g_3'. We can use the MDF formulation,
the SLSQP optimization algorithm
and a maximum number of iterations equal to 100.

.. GENERATED FROM PYTHON SOURCE LINES 80-92

.. code-block:: default

    scenario = create_scenario(
        disciplines,
        formulation="MDF",
        objective_name="y_4",
        maximize_objective=True,
        design_space=design_space,
    )
    scenario.set_differentiation_method()
    for constraint in ["g_1", "g_2", "g_3"]:
        scenario.add_constraint(constraint, "ineq")
    scenario.execute({"algo": "SLSQP", "max_iter": 10})





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

 .. code-block:: none

        INFO - 08:27:57:  
        INFO - 08:27:57: *** Start MDOScenario execution ***
        INFO - 08:27:57: MDOScenario
        INFO - 08:27:57:    Disciplines: SobieskiAerodynamics SobieskiMission SobieskiPropulsion SobieskiStructure
        INFO - 08:27:57:    MDO formulation: MDF
        INFO - 08:27:57: Optimization problem:
        INFO - 08:27:57:    minimize -y_4(x_shared, x_1, x_2, x_3)
        INFO - 08:27:57:    with respect to x_1, x_2, x_3, x_shared
        INFO - 08:27:57:    subject to constraints:
        INFO - 08:27:57:       g_1(x_shared, x_1, x_2, x_3) <= 0.0
        INFO - 08:27:57:       g_2(x_shared, x_1, x_2, x_3) <= 0.0
        INFO - 08:27:57:       g_3(x_shared, x_1, x_2, x_3) <= 0.0
        INFO - 08:27:57:    over the design space:
        INFO - 08:27:57:    +-------------+-------------+-------+-------------+-------+
        INFO - 08:27:57:    | name        | lower_bound | value | upper_bound | type  |
        INFO - 08:27:57:    +-------------+-------------+-------+-------------+-------+
        INFO - 08:27:57:    | x_shared[0] |     0.01    |  0.05 |     0.09    | float |
        INFO - 08:27:57:    | x_shared[1] |    30000    | 45000 |    60000    | float |
        INFO - 08:27:57:    | x_shared[2] |     1.4     |  1.6  |     1.8     | float |
        INFO - 08:27:57:    | x_shared[3] |     2.5     |  5.5  |     8.5     | float |
        INFO - 08:27:57:    | x_shared[4] |      40     |   55  |      70     | float |
        INFO - 08:27:57:    | x_shared[5] |     500     |  1000 |     1500    | float |
        INFO - 08:27:57:    | x_1[0]      |     0.1     |  0.25 |     0.4     | float |
        INFO - 08:27:57:    | x_1[1]      |     0.75    |   1   |     1.25    | float |
        INFO - 08:27:57:    | x_2         |     0.75    |   1   |     1.25    | float |
        INFO - 08:27:57:    | x_3         |     0.1     |  0.5  |      1      | float |
        INFO - 08:27:57:    +-------------+-------------+-------+-------------+-------+
        INFO - 08:27:57: Solving optimization problem with algorithm SLSQP:
        INFO - 08:27:57: ...   0%|          | 0/10 [00:00<?, ?it]
        INFO - 08:27:57: ...  10%|█         | 1/10 [00:00<00:00, 10.06 it/sec, obj=-536]
        INFO - 08:27:58: ...  20%|██        | 2/10 [00:00<00:01,  7.44 it/sec, obj=-2.12e+3]
     WARNING - 08:27:58: MDAJacobi has reached its maximum number of iterations but the normed residual 1.7130677857005655e-05 is still above the tolerance 1e-06.
        INFO - 08:27:58: ...  30%|███       | 3/10 [00:00<00:01,  6.25 it/sec, obj=-3.75e+3]
        INFO - 08:27:58: ...  40%|████      | 4/10 [00:00<00:01,  5.97 it/sec, obj=-3.96e+3]
        INFO - 08:27:58: ...  50%|█████     | 5/10 [00:00<00:00,  5.83 it/sec, obj=-3.96e+3]
        INFO - 08:27:58: Optimization result:
        INFO - 08:27:58:    Optimizer info:
        INFO - 08:27:58:       Status: 8
        INFO - 08:27:58:       Message: Positive directional derivative for linesearch
        INFO - 08:27:58:       Number of calls to the objective function by the optimizer: 6
        INFO - 08:27:58:    Solution:
        INFO - 08:27:58:       The solution is feasible.
        INFO - 08:27:58:       Objective: -3963.408265187933
        INFO - 08:27:58:       Standardized constraints:
        INFO - 08:27:58:          g_1 = [-0.01806104 -0.03334642 -0.04424946 -0.0518346  -0.05732607 -0.13720865
        INFO - 08:27:58:  -0.10279135]
        INFO - 08:27:58:          g_2 = 3.333278582928756e-06
        INFO - 08:27:58:          g_3 = [-7.67181773e-01 -2.32818227e-01  8.30379541e-07 -1.83255000e-01]
        INFO - 08:27:58:       Design space:
        INFO - 08:27:58:       +-------------+-------------+---------------------+-------------+-------+
        INFO - 08:27:58:       | name        | lower_bound |        value        | upper_bound | type  |
        INFO - 08:27:58:       +-------------+-------------+---------------------+-------------+-------+
        INFO - 08:27:58:       | x_shared[0] |     0.01    | 0.06000083331964572 |     0.09    | float |
        INFO - 08:27:58:       | x_shared[1] |    30000    |        60000        |    60000    | float |
        INFO - 08:27:58:       | x_shared[2] |     1.4     |         1.4         |     1.8     | float |
        INFO - 08:27:58:       | x_shared[3] |     2.5     |         2.5         |     8.5     | float |
        INFO - 08:27:58:       | x_shared[4] |      40     |          70         |      70     | float |
        INFO - 08:27:58:       | x_shared[5] |     500     |         1500        |     1500    | float |
        INFO - 08:27:58:       | x_1[0]      |     0.1     |         0.4         |     0.4     | float |
        INFO - 08:27:58:       | x_1[1]      |     0.75    |         0.75        |     1.25    | float |
        INFO - 08:27:58:       | x_2         |     0.75    |         0.75        |     1.25    | float |
        INFO - 08:27:58:       | x_3         |     0.1     |  0.1562448753887276 |      1      | float |
        INFO - 08:27:58:       +-------------+-------------+---------------------+-------------+-------+
        INFO - 08:27:58: *** End MDOScenario execution (time: 0:00:00.973783) ***

    {'max_iter': 10, 'algo': 'SLSQP'}



.. GENERATED FROM PYTHON SOURCE LINES 93-99

Post-process scenario
---------------------
Lastly, we post-process the scenario by means of the :class:`.QuadApprox`
plot which performs a quadratic approximation of a given function
from an optimization history and plot the results as cuts of the
approximation.

.. GENERATED FROM PYTHON SOURCE LINES 101-109

.. tip::

   Each post-processing method requires different inputs and offers a variety
   of customization options. Use the high-level function
   :func:`.get_post_processing_options_schema` to print a table with
   the options for any post-processing algorithm.
   Or refer to our dedicated page:
   :ref:`gen_post_algos`.

.. GENERATED FROM PYTHON SOURCE LINES 111-123

The first plot shows an approximation of the Hessian matrix
:math:`\frac{\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:`\frac{\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:`\frac{\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.

.. GENERATED FROM PYTHON SOURCE LINES 123-126

.. code-block:: default


    scenario.post_process("QuadApprox", function="-y_4", save=False, show=True)




.. 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


    <gemseo.post.quad_approx.QuadApprox object at 0x7f0cb1524100>



.. GENERATED FROM PYTHON SOURCE LINES 127-137

The second plot represents the quadratic approximation of the objective around the
optimal solution : :math:`a_{i}(t)=0.5 (t-x^*_i)^2
\frac{\partial^2 f}{\partial x_i^2} + (t-x^*_i) \frac{\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:`\frac{\partial^2 f}{\partial x_i^2}`.


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

   **Total running time of the script:** (0 minutes 1.826 seconds)


.. _sphx_glr_download_examples_post_process_algorithms_plot_quad_approx.py:

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