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

.. only:: html

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

        Click :ref:`here <sphx_glr_download_examples_mdo_plot_sobieski_use_case.py>`
        to download the full example code

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

.. _sphx_glr_examples_mdo_plot_sobieski_use_case.py:


Application: Sobieski's Super-Sonic Business Jet (MDO)
======================================================
.. _sobieski_mdo:

.. GENERATED FROM PYTHON SOURCE LINES 29-62

This section describes how to setup and solve the MDO problem relative to the
:ref:`Sobieski test case <sobieski_problem>` with |g|.

.. seealso::

   To begin with a more simple MDO problem, and have a detailed description
   of how to plug a test case to |g|, see :ref:`sellar_mdo`.


.. _sobieski_use_case:

Solving with an :ref:`MDF formulation <mdf_formulation>`
--------------------------------------------------------

In this example, we solve the range optimization using the following
:ref:`MDF formulation <mdf_formulation>`:

- The :ref:`MDF formulation <mdf_formulation>` couples all the disciplines
  during the :ref:`mda` at each optimization iteration.
- All the :term:`design variables` are equally treated, concatenated in a
  single vector and given to a single :term:`optimization algorithm` as the
  unknowns of the problem.
- There is no specific :term:`constraint` due to the :ref:`MDF formulation
  <mdf_formulation>`.
- Only the design :term:`constraints` :math:`g\_1`, :math:`g\_2` and
  :math:`g\_3` are added to the problem.
- The :term:`objective function` is the range (the :math:`y\_4` variable in
  the model), computed after the :ref:`mda`.

Imports
-------
All the imports needed for the tutorials are performed here.
Note that some of the imports are related to the Python 2/3 compatibility.

.. GENERATED FROM PYTHON SOURCE LINES 62-80

.. code-block:: default


    from __future__ import division, unicode_literals

    from matplotlib import pyplot as plt

    from gemseo.api import (
        configure_logger,
        create_discipline,
        create_scenario,
        get_all_inputs,
        get_all_outputs,
        get_available_formulations,
    )
    from gemseo.core.jacobian_assembly import JacobianAssembly
    from gemseo.problems.sobieski.core import SobieskiProblem

    configure_logger()





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

 Out:

 .. code-block:: none


    <RootLogger root (INFO)>



.. GENERATED FROM PYTHON SOURCE LINES 81-87

Step 1: Creation of :class:`.MDODiscipline`
-------------------------------------------

To build the scenario, we first instantiate the disciplines. Here, the
disciplines themselves have already been
developed and interfaced with |g| (see :ref:`benchmark_problems`).

.. GENERATED FROM PYTHON SOURCE LINES 87-97

.. code-block:: default


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








.. GENERATED FROM PYTHON SOURCE LINES 98-105

.. tip::

   For the disciplines that are not interfaced with |g|, the |g|'s
   :mod:`~gemseo.api` eases the creation of disciplines without having
   to import them.

   See :ref:`api`.

.. GENERATED FROM PYTHON SOURCE LINES 107-123

Step 2: Creation of :class:`.Scenario`
--------------------------------------

The scenario delegates the creation of the optimization problem to the
:ref:`MDO formulation <mdo_formulations>`.

Therefore, it needs the list of :code:`disciplines`, the names of the formulation,
the name of the objective function and the design space.

- The :code:`design_space` (shown below for reference, as :code:`design_space.txt`)
  defines the unknowns of the optimization problem, and their bounds. It contains
  all the design variables needed by the :ref:`MDF formulation <mdf_formulation>`.
  It can be imported from a text file, or created from scratch with the methods
  :meth:`~gemseo.api.create_design_space` and
  :meth:`~gemseo.algos.design_space.DesignSpace.add_variable`. In this case,
  we will create it directly from the API.

.. GENERATED FROM PYTHON SOURCE LINES 123-124

.. code-block:: default

    design_space = SobieskiProblem().read_design_space()







.. GENERATED FROM PYTHON SOURCE LINES 125-158

.. code::

          vi design_space.txt

          name      lower_bound      value      upper_bound  type
          x_shared      0.01          0.05          0.09     float
          x_shared    30000.0       45000.0       60000.0    float
          x_shared      1.4           1.6           1.8      float
          x_shared      2.5           5.5           8.5      float
          x_shared      40.0          55.0          70.0     float
          x_shared     500.0         1000.0        1500.0    float
          x_1           0.1           0.25          0.4      float
          x_1           0.75          1.0           1.25     float
          x_2           0.75          1.0           1.25     float
          x_3           0.1           0.5           1.0      float
          y_14        24850.0    50606.9741711    77100.0    float
          y_14        -7700.0    7306.20262124    45000.0    float
          y_32         0.235       0.50279625      0.795     float
          y_31         2960.0    6354.32430691    10185.0    float
          y_24          0.44       4.15006276      11.13     float
          y_34          0.44       1.10754577       1.98     float
          y_23         3365.0    12194.2671934    26400.0    float
          y_21        24850.0    50606.9741711    77250.0    float
          y_12        24850.0      50606.9742     77250.0    float
          y_12          0.45          0.95          1.5      float

- The available :ref:`MDO formulations <mdo_formulations>` are located in the
  **gemseo.formulations** package, see :ref:`extending-gemseo` for extending
  GEMSEO with other formulations.
- The :code:`formulation` classname (here, :code:`"MDF"`) shall be passed to
  the scenario to select them.
- The list of available formulations can be obtained by using
  :meth:`~gemseo.api.get_available_formulations`.

.. GENERATED FROM PYTHON SOURCE LINES 158-159

.. code-block:: default

    get_available_formulations()




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

 Out:

 .. code-block:: none


    ['BiLevel', 'DisciplinaryOpt', 'IDF', 'MDF']



.. GENERATED FROM PYTHON SOURCE LINES 160-164

- :math:`y\_4` corresponds to the :code:`objective_name`. This name must be one
  of the disciplines outputs, here the "SobieskiMission" discipline. The list of
  all outputs of the disciplines can be obtained by using
  :meth:`~gemseo.api.get_all_outputs`:

.. GENERATED FROM PYTHON SOURCE LINES 164-166

.. code-block:: default

    get_all_outputs(disciplines)
    get_all_inputs(disciplines)




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

 Out:

 .. code-block:: none


    ['y_23', 'x_shared', 'y_32', 'y_14', 'x_1', 'x_2', 'y_31', 'y_24', 'x_3', 'y_21', 'y_34', 'y_12']



.. GENERATED FROM PYTHON SOURCE LINES 167-170

From these :class:`~gemseo.core.discipline.MDODiscipline`, design space filename,
:ref:`MDO formulation <mdo_formulations>` name and objective function name,
we build the scenario:

.. GENERATED FROM PYTHON SOURCE LINES 170-177

.. code-block:: default

    scenario = create_scenario(
        disciplines,
        formulation="MDF",
        maximize_objective=True,
        objective_name="y_4",
        design_space=design_space,
    )







.. GENERATED FROM PYTHON SOURCE LINES 178-192

The range function (:math:`y\_4`) should be maximized. However, optimizers
minimize functions by default. Which is why, when creating the scenario, the argument
:code:`maximize_objective` shall be set to :code:`True`.

Scenario options
~~~~~~~~~~~~~~~~

We may provide additional options to the scenario:


**Function derivatives.** As analytical disciplinary derivatives are
vailable for Sobieski test-case, they can be used instead of computing
the derivatives with finite-differences or with the complex-step method.
The easiest way to set a method is to let the optimizer determine it:

.. GENERATED FROM PYTHON SOURCE LINES 192-193

.. code-block:: default

    scenario.set_differentiation_method("user")







.. GENERATED FROM PYTHON SOURCE LINES 194-219

The default behavior of the optimizer triggers :term:`finite differences`.
It corresponds to:

.. code::

  scenario.set_differentiation_method("finite_differences",1e-7)

It it also possible to differentiate functions by means of the
:term:`complex step` method:

.. code::

  scenario.set_differentiation_method("complex_step",1e-30j)

Constraints
~~~~~~~~~~~

Similarly to the objective function, the constraints names are a subset
of the disciplines' outputs. They can be obtained by using
:meth:`~gemseo.api.get_all_outputs`.

The formulation has a powerful feature to automatically dispatch the constraints
(:math:`g\_1, g\_2, g\_3`) and plug them to the optimizers depending on
the formulation. To do that, we use the method
:meth:`~gemseo.core.scenario.Scenario.add_constraint`:

.. GENERATED FROM PYTHON SOURCE LINES 220-222

.. code-block:: default

    for constraint in ["g_1", "g_2", "g_3"]:
        scenario.add_constraint(constraint, "ineq")







.. GENERATED FROM PYTHON SOURCE LINES 223-228

Step 3: Execution and visualization of the results
--------------------------------------------------

The algorithm arguments are provided as a dictionary to the execution
method of the scenario:

.. GENERATED FROM PYTHON SOURCE LINES 228-229

.. code-block:: default

    algo_args = {"max_iter": 10, "algo": "SLSQP"}







.. GENERATED FROM PYTHON SOURCE LINES 230-240

.. warning::

   The mandatory arguments are the maximum number of iterations and the algorithm name.
   Any other options of the optimization algorithm can be prescribed through
   the argument :code:`algo_options` with a dictionary, e.g.
   :code:`algo_args =
   {"max_iter": 10, "algo": "SLSQP": "algo_options": {"ftol_rel": 1e-6}}`.
   This list of available algorithm options are detailed here: :ref:`gen_opt_algos`.

The scenario is executed by means of the line:

.. GENERATED FROM PYTHON SOURCE LINES 240-241

.. code-block:: default

    scenario.execute(algo_args)




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

 Out:

 .. code-block:: none

        INFO - 12:59:17:  
        INFO - 12:59:17: *** Start MDO Scenario execution ***
        INFO - 12:59:17: MDOScenario
        INFO - 12:59:17:    Disciplines: SobieskiPropulsion SobieskiAerodynamics SobieskiMission SobieskiStructure
        INFO - 12:59:17:    MDOFormulation: MDF
        INFO - 12:59:17:    Algorithm: SLSQP
        INFO - 12:59:17: Optimization problem:
        INFO - 12:59:17:    Minimize: -y_4(x_shared, x_1, x_2, x_3)
        INFO - 12:59:17:    With respect to: x_shared, x_1, x_2, x_3
        INFO - 12:59:17:    Subject to constraints:
        INFO - 12:59:17:       g_1(x_shared, x_1, x_2, x_3) <= 0.0
        INFO - 12:59:17:       g_2(x_shared, x_1, x_2, x_3) <= 0.0
        INFO - 12:59:17:       g_3(x_shared, x_1, x_2, x_3) <= 0.0
        INFO - 12:59:17: Design space:
        INFO - 12:59:17: +----------+-------------+-------+-------------+-------+
        INFO - 12:59:17: | name     | lower_bound | value | upper_bound | type  |
        INFO - 12:59:17: +----------+-------------+-------+-------------+-------+
        INFO - 12:59:17: | x_shared |     0.01    |  0.05 |     0.09    | float |
        INFO - 12:59:17: | x_shared |    30000    | 45000 |    60000    | float |
        INFO - 12:59:17: | x_shared |     1.4     |  1.6  |     1.8     | float |
        INFO - 12:59:17: | x_shared |     2.5     |  5.5  |     8.5     | float |
        INFO - 12:59:17: | x_shared |      40     |   55  |      70     | float |
        INFO - 12:59:17: | x_shared |     500     |  1000 |     1500    | float |
        INFO - 12:59:17: | x_1      |     0.1     |  0.25 |     0.4     | float |
        INFO - 12:59:17: | x_1      |     0.75    |   1   |     1.25    | float |
        INFO - 12:59:17: | x_2      |     0.75    |   1   |     1.25    | float |
        INFO - 12:59:17: | x_3      |     0.1     |  0.5  |      1      | float |
        INFO - 12:59:17: +----------+-------------+-------+-------------+-------+
        INFO - 12:59:17: Optimization:   0%|          | 0/10 [00:00<?, ?it]
    /home/docs/checkouts/readthedocs.org/user_builds/gemseo/conda/3.2.1/lib/python3.8/site-packages/scipy/sparse/linalg/dsolve/linsolve.py:407: SparseEfficiencyWarning: splu requires CSC matrix format
      warn('splu requires CSC matrix format', SparseEfficiencyWarning)
        INFO - 12:59:17: Optimization:  20%|██        | 2/10 [00:00<00:00, 53.08 it/sec, obj=2.12e+3]
        INFO - 12:59:17: Optimization:  40%|████      | 4/10 [00:00<00:00, 21.28 it/sec, obj=3.97e+3]
        INFO - 12:59:18: Optimization:  50%|█████     | 5/10 [00:00<00:00, 16.60 it/sec, obj=3.96e+3]
        INFO - 12:59:18: Optimization:  60%|██████    | 6/10 [00:00<00:00, 13.61 it/sec, obj=3.96e+3]
        INFO - 12:59:18: Optimization:  70%|███████   | 7/10 [00:00<00:00, 11.54 it/sec, obj=3.96e+3]
        INFO - 12:59:18: Optimization:  90%|█████████ | 9/10 [00:01<00:00,  9.87 it/sec, obj=3.96e+3]
        INFO - 12:59:18: Optimization: 100%|██████████| 10/10 [00:01<00:00,  9.18 it/sec, obj=3.96e+3]
        INFO - 12:59:18: Optimization result:
        INFO - 12:59:18: Objective value = 3963.595455433326
        INFO - 12:59:18: The result is feasible.
        INFO - 12:59:18: Status: None
        INFO - 12:59:18: Optimizer message: Maximum number of iterations reached. GEMSEO Stopped the driver
        INFO - 12:59:18: Number of calls to the objective function by the optimizer: 12
        INFO - 12:59:18: Constraints values:
        INFO - 12:59:18:    g_1 = [-0.01814919 -0.03340982 -0.04429875 -0.05187486 -0.05736009 -0.13720854
        INFO - 12:59:18:  -0.10279146]
        INFO - 12:59:18:    g_2 = 3.236261671801799e-05
        INFO - 12:59:18:    g_3 = [-7.67067574e-01 -2.32932426e-01 -9.19662628e-05 -1.83255000e-01]
        INFO - 12:59:18: Design space:
        INFO - 12:59:18: +----------+-------------+--------------------+-------------+-------+
        INFO - 12:59:18: | name     | lower_bound |       value        | upper_bound | type  |
        INFO - 12:59:18: +----------+-------------+--------------------+-------------+-------+
        INFO - 12:59:18: | x_shared |     0.01    | 0.0600080906541795 |     0.09    | float |
        INFO - 12:59:18: | x_shared |    30000    |       60000        |    60000    | float |
        INFO - 12:59:18: | x_shared |     1.4     |        1.4         |     1.8     | float |
        INFO - 12:59:18: | x_shared |     2.5     |        2.5         |     8.5     | float |
        INFO - 12:59:18: | x_shared |      40     |         70         |      70     | float |
        INFO - 12:59:18: | x_shared |     500     |        1500        |     1500    | float |
        INFO - 12:59:18: | x_1      |     0.1     | 0.3999993439500847 |     0.4     | float |
        INFO - 12:59:18: | x_1      |     0.75    |        0.75        |     1.25    | float |
        INFO - 12:59:18: | x_2      |     0.75    |        0.75        |     1.25    | float |
        INFO - 12:59:18: | x_3      |     0.1     | 0.156230376400943  |      1      | float |
        INFO - 12:59:18: +----------+-------------+--------------------+-------------+-------+
        INFO - 12:59:18: *** MDO Scenario run terminated in 0:00:01.098863 ***

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



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Post-processing options
~~~~~~~~~~~~~~~~~~~~~~~
A whole variety of visualizations may be displayed for both MDO and DOE
scenarios. These features are illustrated on the SSBJ use case in
:ref:`post_processing`.

To visualize the optimization history:

.. GENERATED FROM PYTHON SOURCE LINES 249-252

.. code-block:: default

    scenario.post_process("OptHistoryView", save=False, show=False)
    # Workaround for HTML rendering, instead of ``show=True``
    plt.show()



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


    *

      .. image:: /examples/mdo/images/sphx_glr_plot_sobieski_use_case_001.png
          :alt: Evolution of the optimization variables
          :class: sphx-glr-multi-img

    *

      .. image:: /examples/mdo/images/sphx_glr_plot_sobieski_use_case_002.png
          :alt: Evolution of the objective value
          :class: sphx-glr-multi-img

    *

      .. image:: /examples/mdo/images/sphx_glr_plot_sobieski_use_case_003.png
          :alt: Distance to the optimum
          :class: sphx-glr-multi-img

    *

      .. image:: /examples/mdo/images/sphx_glr_plot_sobieski_use_case_004.png
          :alt: Hessian diagonal approximation
          :class: sphx-glr-multi-img

    *

      .. image:: /examples/mdo/images/sphx_glr_plot_sobieski_use_case_005.png
          :alt: Evolution of the inequality constraints
          :class: sphx-glr-multi-img





.. GENERATED FROM PYTHON SOURCE LINES 253-261

Influence of gradient computation method on performance
-------------------------------------------------------

As mentioned in :ref:`jacobian_assembly`, several methods
are available in order to perform  the gradient computations: classical finite
differences, complex step and :ref:`mda` linearization in direct or adjoint mode.
These modes are automatically selected by |g| to minimize the CPU time. Yet, they
can be forced on demand in each :ref:`mda`:

.. GENERATED FROM PYTHON SOURCE LINES 261-263

.. code-block:: default

    scenario.formulation.mda.linearization_mode = JacobianAssembly.DIRECT_MODE
    scenario.formulation.mda.matrix_type = JacobianAssembly.LINEAR_OPERATOR







.. GENERATED FROM PYTHON SOURCE LINES 264-269

The method used to solve the adjoint or direct linear problem may also be selected.
|g| can either assemble a sparse residual Jacobian matrix of the :ref:`mda` from the
disciplines matrices. This has the advantage that LU factorizations may be stored to
solve multiple right hand sides problems in a cheap way. But this requires
extra memory.

.. GENERATED FROM PYTHON SOURCE LINES 269-271

.. code-block:: default

    scenario.formulation.mda.matrix_type = JacobianAssembly.SPARSE
    scenario.formulation.mda.use_lu_fact = True







.. GENERATED FROM PYTHON SOURCE LINES 272-276

Altenatively, |g| can implicitly create a matrix-vector product operator,
which is sufficient for GMRES-like solvers. It avoids to create an additional
data structure. This can also be mandatory if the disciplines do not provide
full Jacobian matrices but only matrix-vector product operators.

.. GENERATED FROM PYTHON SOURCE LINES 276-277

.. code-block:: default

    scenario.formulation.mda.matrix_type = JacobianAssembly.LINEAR_OPERATOR







.. GENERATED FROM PYTHON SOURCE LINES 278-299

The next table shows the performance of each method for solving the Sobieski use case
with :ref:`MDF <mdf_formulation>` and :ref:`IDF <idf_formulation>` formulations.
Efficiency of linearization is clearly visible has it takes from 10 to 20 times
less CPU time to compute analytic derivatives of an :ref:`mda` compared
to finite difference and complex step.
For :ref:`IDF <idf_formulation>`, improvements are less consequent,
but direct linearization is more than 2.5 times faster than other methods.

.. tabularcolumns:: |l|c|c|

+-----------------------+------------------------------+------------------------------+
|                       |  Execution time (s)                                         |
+  Derivation Method    +------------------------------+------------------------------+
|                       | :ref:`MDF <mdf_formulation>` | :ref:`IDF <idf_formulation>` |
+=======================+==============================+==============================+
| Finite differences    | 8.22                         | 1.93                         |
+-----------------------+------------------------------+------------------------------+
| Complex step          | 18.11                        | 2.07                         |
+-----------------------+------------------------------+------------------------------+
| Linearized (direct)   | 0.90                         | 0.68                         |
+-----------------------+------------------------------+------------------------------+


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

   **Total running time of the script:** ( 0 minutes  2.381 seconds)


.. _sphx_glr_download_examples_mdo_plot_sobieski_use_case.py:


.. only :: html

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



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

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



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

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


.. only:: html

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

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