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

.. only:: html

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

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

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

.. _sphx_glr_auto_numerical_methods_optimization_plot_ego.py:


EfficientGlobalOptimization examples
====================================

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The EGO algorithm [jones1998]_ is an adaptative optimization method based on
kriging.

An initial design of experiment is used to build a first metamodel.
At each iteration a new point that maximizes a criterion is chosen as
optimizer candidate.

The criterion uses a tradeoff between the metamodel value and the conditional
variance.

Then the new point is evaluated using the original model and the metamodel is
relearnt on the extended design of experiments.

.. GENERATED FROM PYTHON SOURCE LINES 22-33

.. code-block:: Python

    from openturns.usecases import branin_function
    from openturns.usecases import ackley_function
    import openturns as ot
    import openturns.viewer as viewer
    from matplotlib import pylab as plt

    ot.RandomGenerator.SetSeed(0)
    ot.ResourceMap.SetAsString("KrigingAlgorithm-LinearAlgebra", "LAPACK")
    ot.Log.Show(ot.Log.NONE)









.. GENERATED FROM PYTHON SOURCE LINES 34-38

Ackley test-case
----------------

We first apply the EGO algorithm on the :ref:`Ackley function<use-case-ackley>`.

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Define the problem
^^^^^^^^^^^^^^^^^^

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The Ackley model is defined in the usecases module in a data class `AckleyModel` :

.. GENERATED FROM PYTHON SOURCE LINES 45-47

.. code-block:: Python

    am = ackley_function.AckleyModel()








.. GENERATED FROM PYTHON SOURCE LINES 48-49

We get the Ackley function :

.. GENERATED FROM PYTHON SOURCE LINES 49-51

.. code-block:: Python

    model = am.model








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We specify the domain of the model :

.. GENERATED FROM PYTHON SOURCE LINES 53-57

.. code-block:: Python

    dim = am.dim
    lowerbound = [-4.0] * dim
    upperbound = [4.0] * dim








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We know that the global minimum is at the center of the domain. It is stored in the `AckleyModel` data class.

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.. code-block:: Python

    xexact = am.x0








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The minimum value attained `fexact` is :

.. GENERATED FROM PYTHON SOURCE LINES 63-66

.. code-block:: Python

    fexact = model(xexact)
    fexact






.. raw:: html

    <div class="output_subarea output_html rendered_html output_result">
    class=Point name=Unnamed dimension=1 values=[4.44089e-16]
    </div>
    <br />
    <br />

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.. code-block:: Python

    graph = model.draw(lowerbound, upperbound, [100] * dim)
    graph.setTitle("Ackley function")
    view = viewer.View(graph)




.. image-sg:: /auto_numerical_methods/optimization/images/sphx_glr_plot_ego_001.png
   :alt: Ackley function
   :srcset: /auto_numerical_methods/optimization/images/sphx_glr_plot_ego_001.png
   :class: sphx-glr-single-img





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We see that the Ackley function has many local minimas. The global minimum, however, is unique and located at the center of the domain.

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Create the initial kriging
^^^^^^^^^^^^^^^^^^^^^^^^^^

Before using the EGO algorithm, we must create an initial kriging.
In order to do this, we must create a design of experiment which fills the space.
In this situation, the :class:`~openturns.LHSExperiment` is a good place to start (but other design of experiments may allow one to better fill the space).
We use a uniform distribution in order to create a LHS design.
The length of the first LHS is set to ten times the problem dimension as recommended in [jones1998]_.

.. GENERATED FROM PYTHON SOURCE LINES 85-94

.. code-block:: Python

    listUniformDistributions = [
        ot.Uniform(lowerbound[i], upperbound[i]) for i in range(dim)
    ]
    distribution = ot.JointDistribution(listUniformDistributions)
    sampleSize = 10 * dim
    experiment = ot.LHSExperiment(distribution, sampleSize)
    inputSample = experiment.generate()
    outputSample = model(inputSample)








.. GENERATED FROM PYTHON SOURCE LINES 95-102

.. code-block:: Python

    graph = ot.Graph(
        "Initial LHS design of experiment - n=%d" % (sampleSize), "$x_0$", "$x_1$", True
    )
    cloud = ot.Cloud(inputSample)
    graph.add(cloud)
    view = viewer.View(graph)




.. image-sg:: /auto_numerical_methods/optimization/images/sphx_glr_plot_ego_002.png
   :alt: Initial LHS design of experiment - n=20
   :srcset: /auto_numerical_methods/optimization/images/sphx_glr_plot_ego_002.png
   :class: sphx-glr-single-img





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We now create the kriging metamodel.
We selected the :class:`~openturns.MaternModel` covariance model with a constant basis as recommended in [leriche2021]_.

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.. code-block:: Python

    covarianceModel = ot.MaternModel([1.0] * dim, [0.5], 2.5)
    basis = ot.ConstantBasisFactory(dim).build()
    kriging = ot.KrigingAlgorithm(inputSample, outputSample, covarianceModel, basis)
    kriging.run()








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Create the optimization problem
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

We finally create the :class:`~openturns.OptimizationProblem` and solve it with class:`~openturns.EfficientGlobalOptimization`.

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.. code-block:: Python

    problem = ot.OptimizationProblem()
    problem.setObjective(model)
    bounds = ot.Interval(lowerbound, upperbound)
    problem.setBounds(bounds)








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In order to show the various options, we configure them all, even if most could be left to default settings in this case.

The most important method is :class:`~openturns.EfficientGlobalOptimization.setMaximumCallsNumber` which limits the number of iterations that the algorithm can perform.
In the Ackley example, we choose to perform 30 iterations of the algorithm.

.. GENERATED FROM PYTHON SOURCE LINES 131-136

.. code-block:: Python

    algo = ot.EfficientGlobalOptimization(problem, kriging.getResult())
    algo.setMaximumCallsNumber(30)
    algo.run()
    result = algo.getResult()








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.. code-block:: Python

    result.getIterationNumber()





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

 .. code-block:: none


    30



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.. code-block:: Python

    result.getOptimalPoint()






.. raw:: html

    <div class="output_subarea output_html rendered_html output_result">
    class=Point name=Unnamed dimension=2 values=[-0.0062633,-0.0052574]
    </div>
    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 143-145

.. code-block:: Python

    result.getOptimalValue()






.. raw:: html

    <div class="output_subarea output_html rendered_html output_result">
    class=Point name=Unnamed dimension=1 values=[0.0249089]
    </div>
    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 146-148

.. code-block:: Python

    fexact






.. raw:: html

    <div class="output_subarea output_html rendered_html output_result">
    class=Point name=Unnamed dimension=1 values=[4.44089e-16]
    </div>
    <br />
    <br />

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Compared to the minimum function value, we see that the EGO algorithm
provides solution which is accurate.
Indeed, the optimal point is in the neighbourhood of the exact solution,
and this is quite an impressive success given the limited amount of function
evaluations: only 20 evaluations for the initial DOE and 30 iterations of
the EGO algorithm, for a total equal to 50 function evaluations.

.. GENERATED FROM PYTHON SOURCE LINES 157-162

.. code-block:: Python

    graph = result.drawOptimalValueHistory()
    optimum_curve = ot.Curve(ot.Sample([[0, fexact[0]], [29, fexact[0]]]))
    graph.add(optimum_curve)
    view = viewer.View(graph)




.. image-sg:: /auto_numerical_methods/optimization/images/sphx_glr_plot_ego_003.png
   :alt: Optimal value history
   :srcset: /auto_numerical_methods/optimization/images/sphx_glr_plot_ego_003.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 163-165

.. code-block:: Python

    inputHistory = result.getInputSample()








.. GENERATED FROM PYTHON SOURCE LINES 166-178

.. code-block:: Python

    graph = model.draw(lowerbound, upperbound, [100] * dim)
    graph.setTitle("Ackley function")
    cloud = ot.Cloud(inputSample, "initial")
    cloud.setPointStyle("bullet")
    cloud.setColor("black")
    graph.add(cloud)
    cloud = ot.Cloud(inputHistory, "solution")
    cloud.setPointStyle("diamond")
    cloud.setColor("forestgreen")
    graph.add(cloud)
    view = viewer.View(graph)




.. image-sg:: /auto_numerical_methods/optimization/images/sphx_glr_plot_ego_004.png
   :alt: Ackley function
   :srcset: /auto_numerical_methods/optimization/images/sphx_glr_plot_ego_004.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 179-181

We see that the initial (black) points are dispersed in the whole domain,
while the solution points are much closer to the solution.

.. GENERATED FROM PYTHON SOURCE LINES 183-187

Branin test-case
----------------

We now take a look at the :ref:`Branin-Hoo<use-case-branin>` function.

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Define the problem
^^^^^^^^^^^^^^^^^^

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The Branin model is defined in the usecases module in a data class `BraninModel` :

.. GENERATED FROM PYTHON SOURCE LINES 194-196

.. code-block:: Python

    bm = branin_function.BraninModel()








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We load the dimension,

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.. code-block:: Python

    dim = bm.dim








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the domain boundaries,

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.. code-block:: Python

    lowerbound = bm.lowerbound
    upperbound = bm.upperbound








.. GENERATED FROM PYTHON SOURCE LINES 206-207

and we load the model function and its noise :

.. GENERATED FROM PYTHON SOURCE LINES 207-210

.. code-block:: Python

    objectiveFunction = bm.model
    noise = bm.noiseModel








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We build a sample out of the three minima :

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.. code-block:: Python

    xexact = ot.Sample([bm.xexact1, bm.xexact2, bm.xexact3])








.. GENERATED FROM PYTHON SOURCE LINES 215-216

The minimum value attained `fexact` is :

.. GENERATED FROM PYTHON SOURCE LINES 216-219

.. code-block:: Python

    fexact = objectiveFunction(xexact)
    fexact






.. raw:: html

    <div class="output_subarea output_html rendered_html output_result">
    <table>
      <tr><td></td><th>y0</th></tr>
      <tr><th>0</th><td>-1.04741</td></tr>
      <tr><th>1</th><td>-1.04741</td></tr>
      <tr><th>2</th><td>-1.04741</td></tr>
    </table>
    </div>
    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 220-224

.. code-block:: Python

    graph = objectiveFunction.draw(lowerbound, upperbound, [100] * dim)
    graph.setTitle("Branin function")
    view = viewer.View(graph)




.. image-sg:: /auto_numerical_methods/optimization/images/sphx_glr_plot_ego_005.png
   :alt: Branin function
   :srcset: /auto_numerical_methods/optimization/images/sphx_glr_plot_ego_005.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 225-226

The Branin function has three local minimas.

.. GENERATED FROM PYTHON SOURCE LINES 228-230

Create the initial kriging
^^^^^^^^^^^^^^^^^^^^^^^^^^

.. GENERATED FROM PYTHON SOURCE LINES 232-239

.. code-block:: Python

    distribution = ot.JointDistribution([ot.Uniform(0.0, 1.0)] * dim)
    sampleSize = 10 * dim
    experiment = ot.LHSExperiment(distribution, sampleSize)
    inputSample = experiment.generate()
    outputSample = objectiveFunction(inputSample)
    noiseSample = noise(inputSample)








.. GENERATED FROM PYTHON SOURCE LINES 240-247

.. code-block:: Python

    graph = ot.Graph(
        "Initial LHS design of experiment - n=%d" % (sampleSize), "$x_0$", "$x_1$", True
    )
    cloud = ot.Cloud(inputSample)
    graph.add(cloud)
    view = viewer.View(graph)




.. image-sg:: /auto_numerical_methods/optimization/images/sphx_glr_plot_ego_006.png
   :alt: Initial LHS design of experiment - n=20
   :srcset: /auto_numerical_methods/optimization/images/sphx_glr_plot_ego_006.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 248-252

.. code-block:: Python

    covarianceModel = ot.MaternModel([1.0] * dim, [0.5], 2.5)
    basis = ot.ConstantBasisFactory(dim).build()
    kriging = ot.KrigingAlgorithm(inputSample, outputSample, covarianceModel, basis)








.. GENERATED FROM PYTHON SOURCE LINES 253-256

.. code-block:: Python

    kriging.setNoise([x[0] for x in noiseSample])
    kriging.run()








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Create and solve the problem
^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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We define the problem :

.. GENERATED FROM PYTHON SOURCE LINES 262-267

.. code-block:: Python

    problem = ot.OptimizationProblem()
    problem.setObjective(objectiveFunction)
    bounds = ot.Interval(lowerbound, upperbound)
    problem.setBounds(bounds)








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We configure the algorithm, with the model noise:

.. GENERATED FROM PYTHON SOURCE LINES 269-272

.. code-block:: Python

    algo = ot.EfficientGlobalOptimization(problem, kriging.getResult(), noise)
    algo.setMaximumCallsNumber(30)








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We run the algorithm and get the result:

.. GENERATED FROM PYTHON SOURCE LINES 274-277

.. code-block:: Python

    algo.run()
    result = algo.getResult()








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.. code-block:: Python

    result.getIterationNumber()





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

 .. code-block:: none


    30



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.. code-block:: Python

    result.getOptimalPoint()






.. raw:: html

    <div class="output_subarea output_html rendered_html output_result">
    class=Point name=Unnamed dimension=2 values=[0.528177,0.167468]
    </div>
    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 284-286

.. code-block:: Python

    result.getOptimalValue()






.. raw:: html

    <div class="output_subarea output_html rendered_html output_result">
    class=Point name=Unnamed dimension=1 values=[-1.04293]
    </div>
    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 287-289

.. code-block:: Python

    fexact






.. raw:: html

    <div class="output_subarea output_html rendered_html output_result">
    <table>
      <tr><td></td><th>y0</th></tr>
      <tr><th>0</th><td>-1.04741</td></tr>
      <tr><th>1</th><td>-1.04741</td></tr>
      <tr><th>2</th><td>-1.04741</td></tr>
    </table>
    </div>
    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 290-292

.. code-block:: Python

    inputHistory = result.getInputSample()








.. GENERATED FROM PYTHON SOURCE LINES 293-305

.. code-block:: Python

    graph = objectiveFunction.draw(lowerbound, upperbound, [100] * dim)
    graph.setTitle("Branin function")
    cloud = ot.Cloud(inputSample, "initial")
    cloud.setPointStyle("bullet")
    cloud.setColor("black")
    graph.add(cloud)
    cloud = ot.Cloud(inputHistory, "solution")
    cloud.setPointStyle("diamond")
    cloud.setColor("forestgreen")
    graph.add(cloud)
    view = viewer.View(graph)




.. image-sg:: /auto_numerical_methods/optimization/images/sphx_glr_plot_ego_007.png
   :alt: Branin function
   :srcset: /auto_numerical_methods/optimization/images/sphx_glr_plot_ego_007.png
   :class: sphx-glr-single-img





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We see that the EGO algorithm reached the different optima locations.

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.. code-block:: Python

    graph = result.drawOptimalValueHistory()
    optimum_curve = ot.Curve(ot.Sample([[0, fexact[0][0]], [29, fexact[0][0]]]))
    graph.add(optimum_curve)
    view = viewer.View(graph, axes_kw={"xticks": range(0, result.getIterationNumber(), 5)})
    plt.show()




.. image-sg:: /auto_numerical_methods/optimization/images/sphx_glr_plot_ego_008.png
   :alt: Optimal value history
   :srcset: /auto_numerical_methods/optimization/images/sphx_glr_plot_ego_008.png
   :class: sphx-glr-single-img





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Reset default settings

.. GENERATED FROM PYTHON SOURCE LINES 317-318

.. code-block:: Python

    ot.ResourceMap.Reload()








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

   **Total running time of the script:** (0 minutes 2.282 seconds)


.. _sphx_glr_download_auto_numerical_methods_optimization_plot_ego.py:

.. only:: html

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

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

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

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

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

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

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