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.. "tutorials_sg/mdo/plot_sellar.py"
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.. only:: html

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

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

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

.. _sphx_glr_tutorials_sg_mdo_plot_sellar.py:


A from scratch example on the Sellar problem
============================================
.. _sellar_from_scratch:

.. GENERATED FROM PYTHON SOURCE LINES 29-37

Introduction
------------
In this example,
we will create an MDO scenario based on the Sellar's problem from scratch.
Contrary to the :ref:`|g| in ten minutes tutorial <gemseo_10min>`,
all the disciplines will be implemented from scratch
by sub-classing the :class:`.MDODiscipline` class
for each discipline of the Sellar problem.

.. GENERATED FROM PYTHON SOURCE LINES 39-44

The Sellar problem
------------------
We will consider in this example the Sellar problem:

.. include:: /tutorials/_description/sellar_problem_definition.inc

.. GENERATED FROM PYTHON SOURCE LINES 46-50

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 50-64

.. code-block:: default


    from __future__ import division, unicode_literals

    from math import exp

    from numpy import array, ones

    from gemseo.algos.design_space import DesignSpace
    from gemseo.api import configure_logger, create_scenario
    from gemseo.core.discipline import MDODiscipline

    configure_logger()






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

 Out:

 .. code-block:: none


    <RootLogger root (INFO)>



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Create the disciplinary classes
-------------------------------
In this section,
we define the Sellar disciplines by sub-classing the :class:`.MDODiscipline` class.
For each class,
the constructor and the _run method are overriden:

- In the constructor,
  the input and output grammar are created.
  They define which inputs and outputs variables are allowed
  at the discipline execution.
  The default inputs are also defined,
  in case of the user does not provide them at the discipline execution.
- In the _run method is implemented the concrete computation of the discipline.
  The inputs data are fetch
  by using the :meth:`.MDODiscipline.get_inputs_by_name` method.
  The returned NumPy arrays can then be used to compute the output values.
  They can then be stored in the :attr:`!MDODiscipline.local_data` dictionary.
  If the discipline execution is successful.

Note that we do not define the Jacobians in the disciplines.
In this example,
we will approximate the derivatives using the finite differences method.

Create the SellarSystem class
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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.. code-block:: default



    class SellarSystem(MDODiscipline):
        def __init__(self):
            super(SellarSystem, self).__init__()
            # Initialize the grammars to define inputs and outputs
            self.input_grammar.initialize_from_data_names(["x", "z", "y_1", "y_2"])
            self.output_grammar.initialize_from_data_names(["obj", "c_1", "c_2"])
            # Default inputs define what data to use when the inputs are not
            # provided to the execute method
            self.default_inputs = {
                "x": ones(1),
                "z": array([4.0, 3.0]),
                "y_1": ones(1),
                "y_2": ones(1),
            }

        def _run(self):
            # The run method defines what happens at execution
            # ie how outputs are computed from inputs
            x, z, y_1, y_2 = self.get_inputs_by_name(["x", "z", "y_1", "y_2"])
            # The ouputs are stored here
            self.local_data["obj"] = array([x[0] ** 2 + z[1] + y_1[0] ** 2 + exp(-y_2[0])])
            self.local_data["c_1"] = array([3.16 - y_1[0] ** 2])
            self.local_data["c_2"] = array([y_2[0] - 24.0])









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Create the Sellar1 class
^^^^^^^^^^^^^^^^^^^^^^^^

.. GENERATED FROM PYTHON SOURCE LINES 121-142

.. code-block:: default



    class Sellar1(MDODiscipline):
        def __init__(self):
            super(Sellar1, self).__init__()
            self.input_grammar.initialize_from_data_names(["x", "z", "y_2"])
            self.output_grammar.initialize_from_data_names(["y_1"])
            self.default_inputs = {
                "x": ones(1),
                "z": array([4.0, 3.0]),
                "y_1": ones(1),
                "y_2": ones(1),
            }

        def _run(self):
            x, z, y_2 = self.get_inputs_by_name(["x", "z", "y_2"])
            self.local_data["y_1"] = array(
                [(z[0] ** 2 + z[1] + x[0] - 0.2 * y_2[0]) ** 0.5]
            )









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Create the Sellar2 class
^^^^^^^^^^^^^^^^^^^^^^^^

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.. code-block:: default



    class Sellar2(MDODiscipline):
        def __init__(self):
            super(Sellar2, self).__init__()
            self.input_grammar.initialize_from_data_names(["z", "y_1"])
            self.output_grammar.initialize_from_data_names(["y_2"])
            self.default_inputs = {
                "x": ones(1),
                "z": array([4.0, 3.0]),
                "y_1": ones(1),
                "y_2": ones(1),
            }

        def _run(self):
            z, y_1 = self.get_inputs_by_name(["z", "y_1"])
            self.local_data["y_2"] = array([abs(y_1[0]) + z[0] + z[1]])









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Create and execute the scenario
-------------------------------

Instantiate disciplines
^^^^^^^^^^^^^^^^^^^^^^^
We can now instantiate the disciplines
and store the instances in a list which will be used below.

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.. code-block:: default


    disciplines = [Sellar1(), Sellar2(), SellarSystem()]








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Create the design space
^^^^^^^^^^^^^^^^^^^^^^^
In this section,
we define the design space which will be used for the creation of the MDOScenario.
Note that the coupling variables are defined in the design space.
Indeed,
as we are going to select the IDF formulation to solve the MDO scenario,
the coupling variables will be unknowns of the optimization problem
and consequently they have to be included in the design space.
Conversely,
it would not have been necessary to include them
if we aimed to select an MDF formulation.

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.. code-block:: default


    design_space = DesignSpace()
    design_space.add_variable("x", 1, l_b=0.0, u_b=10.0, value=ones(1))
    design_space.add_variable(
        "z", 2, l_b=(-10, 0.0), u_b=(10.0, 10.0), value=array([4.0, 3.0])
    )
    design_space.add_variable("y_1", 1, l_b=-100.0, u_b=100.0, value=ones(1))
    design_space.add_variable("y_2", 1, l_b=-100.0, u_b=100.0, value=ones(1))








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Create the scenario
^^^^^^^^^^^^^^^^^^^
In this section,
we build the MDO scenario which links the disciplines with the formulation,
the design space and the objective function.

.. GENERATED FROM PYTHON SOURCE LINES 203-207

.. code-block:: default

    scenario = create_scenario(
        disciplines, formulation="IDF", objective_name="obj", design_space=design_space
    )








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Add the constraints
^^^^^^^^^^^^^^^^^^^
Then,
we have to set the design constraints

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.. code-block:: default

    scenario.add_constraint("c_1", "ineq")
    scenario.add_constraint("c_2", "ineq")








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As previously mentioned,
we are going to use finite differences to approximate the derivatives
since the disciplines do not provide them.

.. GENERATED FROM PYTHON SOURCE LINES 219-221

.. code-block:: default

    scenario.set_differentiation_method("finite_differences", 1e-6)








.. GENERATED FROM PYTHON SOURCE LINES 222-228

Execute the scenario
^^^^^^^^^^^^^^^^^^^^
Then,
we execute the MDO scenario with the inputs of the MDO scenario as a dictionary.
In this example,
the gradient-based `SLSQP` optimizer is selected, with 10 iterations at maximum:

.. GENERATED FROM PYTHON SOURCE LINES 228-230

.. code-block:: default

    scenario.execute(input_data={"max_iter": 10, "algo": "SLSQP"})





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

 Out:

 .. code-block:: none

        INFO - 09:26:09:  
        INFO - 09:26:09: *** Start MDO Scenario execution ***
        INFO - 09:26:09: MDOScenario
        INFO - 09:26:09:    Disciplines: Sellar1 Sellar2 SellarSystem
        INFO - 09:26:09:    MDOFormulation: IDF
        INFO - 09:26:09:    Algorithm: SLSQP
        INFO - 09:26:09: Optimization problem:
        INFO - 09:26:09:    Minimize: obj(x, z, y_1, y_2)
        INFO - 09:26:09:    With respect to: x, z, y_1, y_2
        INFO - 09:26:09:    Subject to constraints:
        INFO - 09:26:09:       c_1(x, z, y_1, y_2) <= 0.0
        INFO - 09:26:09:       c_2(x, z, y_1, y_2) <= 0.0
        INFO - 09:26:09:       y_1(x, z, y_2): y_1(x, z, y_2) - y_1 == 0.0
        INFO - 09:26:09:       y_2(z, y_1): y_2(z, y_1) - y_2 == 0.0
        INFO - 09:26:09: Design Space:
        INFO - 09:26:09: +------+-------------+-------+-------------+-------+
        INFO - 09:26:09: | name | lower_bound | value | upper_bound | type  |
        INFO - 09:26:09: +------+-------------+-------+-------------+-------+
        INFO - 09:26:09: | x    |      0      |   1   |      10     | float |
        INFO - 09:26:09: | z    |     -10     |   4   |      10     | float |
        INFO - 09:26:09: | z    |      0      |   3   |      10     | float |
        INFO - 09:26:09: | y_1  |     -100    |   1   |     100     | float |
        INFO - 09:26:09: | y_2  |     -100    |   1   |     100     | float |
        INFO - 09:26:09: +------+-------------+-------+-------------+-------+
        INFO - 09:26:09: Optimization:   0%|          | 0/10 [00:00<?, ?it]
        INFO - 09:26:09: Optimization:  80%|████████  | 8/10 [00:00<00:00, 93.24 it/sec, obj=3.18]
        INFO - 09:26:09: Optimization: 100%|██████████| 10/10 [00:00<00:00, 85.19 it/sec, obj=3.17]
        INFO - 09:26:09: Optimization result:
        INFO - 09:26:09: Objective value = 3.1833939516378016
        INFO - 09:26:09: The result is feasible.
        INFO - 09:26:09: Status: None
        INFO - 09:26:09: Optimizer message: Maximum number of iterations reached. GEMSEO Stopped the driver
        INFO - 09:26:09: Number of calls to the objective function by the optimizer: 12
        INFO - 09:26:09: Constraints values w.r.t. 0: 
        INFO - 09:26:09:    c_1 = 2.849720459607852e-12
        INFO - 09:26:09:    c_2 = -20.244722233075365
        INFO - 09:26:09:    y_1 = [1.62536651e-15]
        INFO - 09:26:09:    y_2 = [1.44773082e-15]
        INFO - 09:26:09: Design Space:
        INFO - 09:26:09: +------+-------------+-------------------+-------------+-------+
        INFO - 09:26:09: | name | lower_bound |       value       | upper_bound | type  |
        INFO - 09:26:09: +------+-------------+-------------------+-------------+-------+
        INFO - 09:26:09: | x    |      0      |         0         |      10     | float |
        INFO - 09:26:09: | z    |     -10     | 1.977638883462609 |      10     | float |
        INFO - 09:26:09: | z    |      0      |         0         |      10     | float |
        INFO - 09:26:09: | y_1  |     -100    | 1.777638883462316 |     100     | float |
        INFO - 09:26:09: | y_2  |     -100    | 3.755277766924635 |     100     | float |
        INFO - 09:26:09: +------+-------------+-------------------+-------------+-------+
        INFO - 09:26:09: *** MDO Scenario run terminated in 0:00:00.130688 ***

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



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Post-process the scenario
^^^^^^^^^^^^^^^^^^^^^^^^^
Finally,
we can generate plots of the optimization history:

.. GENERATED FROM PYTHON SOURCE LINES 235-236

.. code-block:: default

    scenario.post_process("OptHistoryView", save=False, show=True)




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

 Out:

 .. code-block:: none


    <gemseo.post.opt_history_view.OptHistoryView object at 0x7f61b4b4d220>




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

   **Total running time of the script:** ( 0 minutes  0.520 seconds)


.. _sphx_glr_download_tutorials_sg_mdo_plot_sellar.py:


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