.. 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 (Jones, 1998) 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 experiment.

.. GENERATED FROM PYTHON SOURCE LINES 21-32

.. 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 33-37

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 44-46

.. code-block:: Python

    am = ackley_function.AckleyModel()








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We get the Ackley function :

.. GENERATED FROM PYTHON SOURCE LINES 48-50

.. code-block:: Python

    model = am.model








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We specify the domain of the model :

.. GENERATED FROM PYTHON SOURCE LINES 52-56

.. 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.

.. GENERATED FROM PYTHON SOURCE LINES 58-60

.. code-block:: Python

    xexact = am.x0








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The minimum value attained `fexact` is :

.. GENERATED FROM PYTHON SOURCE LINES 62-65

.. 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 />

.. GENERATED FROM PYTHON SOURCE LINES 66-70

.. 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 `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 with 50 points.

.. GENERATED FROM PYTHON SOURCE LINES 83-92

.. code-block:: Python

    listUniformDistributions = [
        ot.Uniform(lowerbound[i], upperbound[i]) for i in range(dim)
    ]
    distribution = ot.ComposedDistribution(listUniformDistributions)
    sampleSize = 50
    experiment = ot.LHSExperiment(distribution, sampleSize)
    inputSample = experiment.generate()
    outputSample = model(inputSample)








.. GENERATED FROM PYTHON SOURCE LINES 93-100

.. 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=50
   :srcset: /auto_numerical_methods/optimization/images/sphx_glr_plot_ego_002.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 101-103

We now create the kriging metamodel. We selected the `SquaredExponential` covariance model with a constant basis (the `MaternModel` may perform better in this case).
We use default settings (1.0) for the scale parameters of the covariance model, but configure the amplitude to 0.1, which better corresponds to the properties of the Ackley function.

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.. code-block:: Python

    covarianceModel = ot.SquaredExponential([1.0] * dim, [0.5])
    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 `OptimizationProblem` and solve it with `EfficientGlobalOptimization`.

.. GENERATED FROM PYTHON SOURCE LINES 117-122

.. code-block:: Python

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








.. GENERATED FROM PYTHON SOURCE LINES 123-127

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 `setMaximumEvaluationNumber` which limits the number of iterations that the algorithm can perform.
In the Ackley example, we choose to perform 10 iterations of the algorithm.

.. GENERATED FROM PYTHON SOURCE LINES 129-134

.. code-block:: Python

    algo = ot.EfficientGlobalOptimization(problem, kriging.getResult())
    algo.setMaximumEvaluationNumber(10)
    algo.run()
    result = algo.getResult()








.. GENERATED FROM PYTHON SOURCE LINES 135-137

.. code-block:: Python

    result.getIterationNumber()





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

 .. code-block:: none


    10



.. GENERATED FROM PYTHON SOURCE LINES 138-140

.. code-block:: Python

    result.getOptimalPoint()






.. raw:: html

    <div class="output_subarea output_html rendered_html output_result">
    class=Point name=Unnamed dimension=2 values=[0.0119045,0.17423]
    </div>
    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 141-143

.. code-block:: Python

    result.getOptimalValue()






.. raw:: html

    <div class="output_subarea output_html rendered_html output_result">
    class=Point name=Unnamed dimension=1 values=[1.13576]
    </div>
    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 144-146

.. 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 />

.. GENERATED FROM PYTHON SOURCE LINES 147-153

Compared to the minimum function value, we see that the EGO algorithm
provides solution which is not very accurate.
However, 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 60 evaluations for the initial DOE and 10 iterations of
the EGO algorithm, for a total equal to 70 function evaluations.

.. GENERATED FROM PYTHON SOURCE LINES 155-158

.. code-block:: Python

    graph = result.drawOptimalValueHistory()
    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 159-161

.. code-block:: Python

    inputHistory = result.getInputSample()








.. GENERATED FROM PYTHON SOURCE LINES 162-174

.. 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 175-180

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

However, the final solution produced by the EGO algorithm is not very accurate.
This is why we finalize the process by adding a local optimization step.

.. GENERATED FROM PYTHON SOURCE LINES 182-187

.. code-block:: Python

    algo2 = ot.NLopt(problem, "LD_LBFGS")
    algo2.setStartingPoint(result.getOptimalPoint())
    algo2.run()
    result = algo2.getResult()








.. GENERATED FROM PYTHON SOURCE LINES 188-190

.. code-block:: Python

    result.getOptimalPoint()






.. raw:: html

    <div class="output_subarea output_html rendered_html output_result">
    class=Point name=Unnamed dimension=2 values=[1.0076e-07,-1.09416e-06]
    </div>
    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 191-192

The corrected solution is now extremely accurate.

.. GENERATED FROM PYTHON SOURCE LINES 194-197

.. code-block:: Python

    graph = result.drawOptimalValueHistory()
    view = viewer.View(graph)




.. image-sg:: /auto_numerical_methods/optimization/images/sphx_glr_plot_ego_005.png
   :alt: Optimal value history
   :srcset: /auto_numerical_methods/optimization/images/sphx_glr_plot_ego_005.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 198-202

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 209-211

.. code-block:: Python

    bm = branin_function.BraninModel()








.. GENERATED FROM PYTHON SOURCE LINES 212-213

We load the dimension,

.. GENERATED FROM PYTHON SOURCE LINES 213-215

.. code-block:: Python

    dim = bm.dim








.. GENERATED FROM PYTHON SOURCE LINES 216-217

the domain boundaries,

.. GENERATED FROM PYTHON SOURCE LINES 217-220

.. code-block:: Python

    lowerbound = bm.lowerbound
    upperbound = bm.upperbound








.. GENERATED FROM PYTHON SOURCE LINES 221-222

and we load the model function and its noise :

.. GENERATED FROM PYTHON SOURCE LINES 222-225

.. code-block:: Python

    objectiveFunction = bm.model
    noise = bm.noiseModel








.. GENERATED FROM PYTHON SOURCE LINES 226-227

We build a sample out of the three minima :

.. GENERATED FROM PYTHON SOURCE LINES 227-229

.. code-block:: Python

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








.. GENERATED FROM PYTHON SOURCE LINES 230-231

The minimum value attained `fexact` is :

.. GENERATED FROM PYTHON SOURCE LINES 231-234

.. 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 235-239

.. 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_006.png
   :alt: Branin function
   :srcset: /auto_numerical_methods/optimization/images/sphx_glr_plot_ego_006.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 240-241

The Branin function has three local minimas.

.. GENERATED FROM PYTHON SOURCE LINES 243-245

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

.. GENERATED FROM PYTHON SOURCE LINES 247-254

.. code-block:: Python

    distribution = ot.ComposedDistribution([ot.Uniform(0.0, 1.0)] * dim)
    sampleSize = 50
    experiment = ot.LHSExperiment(distribution, sampleSize)
    inputSample = experiment.generate()
    outputSample = objectiveFunction(inputSample)
    noiseSample = noise(inputSample)








.. GENERATED FROM PYTHON SOURCE LINES 255-262

.. 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_007.png
   :alt: Initial LHS design of experiment - n=50
   :srcset: /auto_numerical_methods/optimization/images/sphx_glr_plot_ego_007.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 263-267

.. code-block:: Python

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








.. GENERATED FROM PYTHON SOURCE LINES 268-271

.. code-block:: Python

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








.. GENERATED FROM PYTHON SOURCE LINES 272-274

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

.. GENERATED FROM PYTHON SOURCE LINES 276-277

We define the problem :

.. GENERATED FROM PYTHON SOURCE LINES 277-282

.. code-block:: Python

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








.. GENERATED FROM PYTHON SOURCE LINES 283-284

We configure the algorithm, with the model noise:

.. GENERATED FROM PYTHON SOURCE LINES 284-287

.. code-block:: Python

    algo = ot.EfficientGlobalOptimization(problem, kriging.getResult(), noise)
    algo.setMaximumEvaluationNumber(20)








.. GENERATED FROM PYTHON SOURCE LINES 288-289

We run the algorithm and get the result:

.. GENERATED FROM PYTHON SOURCE LINES 289-292

.. 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


    20



.. GENERATED FROM PYTHON SOURCE LINES 296-298

.. code-block:: Python

    result.getOptimalPoint()






.. raw:: html

    <div class="output_subarea output_html rendered_html output_result">
    class=Point name=Unnamed dimension=2 values=[0.547929,0.166692]
    </div>
    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 299-301

.. code-block:: Python

    result.getOptimalValue()






.. raw:: html

    <div class="output_subarea output_html rendered_html output_result">
    class=Point name=Unnamed dimension=1 values=[-1.04529]
    </div>
    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 302-304

.. 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 305-307

.. code-block:: Python

    inputHistory = result.getInputSample()








.. GENERATED FROM PYTHON SOURCE LINES 308-320

.. 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_008.png
   :alt: Branin function
   :srcset: /auto_numerical_methods/optimization/images/sphx_glr_plot_ego_008.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 321-323

We see that the EGO algorithm found the second local minimum.
Given the limited number of function evaluations, the other local minimas have not been explored.

.. GENERATED FROM PYTHON SOURCE LINES 325-330

.. code-block:: Python

    graph = result.drawOptimalValueHistory()
    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_009.png
   :alt: Optimal value history
   :srcset: /auto_numerical_methods/optimization/images/sphx_glr_plot_ego_009.png
   :class: sphx-glr-single-img





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Reset default settings

.. GENERATED FROM PYTHON SOURCE LINES 332-333

.. code-block:: Python

    ot.ResourceMap.Reload()








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

   **Total running time of the script:** (0 minutes 2.444 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>`