.. 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
====================================

.. GENERATED FROM PYTHON SOURCE LINES 6-18

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:: default

    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>`.

.. GENERATED FROM PYTHON SOURCE LINES 39-41

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

.. GENERATED FROM PYTHON SOURCE LINES 43-44

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

.. GENERATED FROM PYTHON SOURCE LINES 44-46

.. code-block:: default

    am = ackley_function.AckleyModel()








.. GENERATED FROM PYTHON SOURCE LINES 47-48

We get the Ackley function :

.. GENERATED FROM PYTHON SOURCE LINES 48-50

.. code-block:: default

    model = am.model








.. GENERATED FROM PYTHON SOURCE LINES 51-52

We specify the domain of the model :

.. GENERATED FROM PYTHON SOURCE LINES 52-56

.. code-block:: default

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








.. GENERATED FROM PYTHON SOURCE LINES 57-58

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:: default

    xexact = am.x0








.. GENERATED FROM PYTHON SOURCE LINES 61-62

The minimum value attained `fexact` is :

.. GENERATED FROM PYTHON SOURCE LINES 62-65

.. code-block:: default

    fexact = model(xexact)
    fexact






.. raw:: html

    <div class="output_subarea output_html rendered_html output_result">
    <p>[4.44089e-16]</p>
    </div>
    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 66-70

.. code-block:: default

    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





.. GENERATED FROM PYTHON SOURCE LINES 71-72

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

.. GENERATED FROM PYTHON SOURCE LINES 74-78

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 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 80-89

.. code-block:: default

    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 90-97

.. code-block:: default

    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 98-99

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.

.. GENERATED FROM PYTHON SOURCE LINES 101-106

.. code-block:: default

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








.. GENERATED FROM PYTHON SOURCE LINES 107-111

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

We finally create the `OptimizationProblem` and solve it with `EfficientGlobalOptimization`.

.. GENERATED FROM PYTHON SOURCE LINES 113-118

.. code-block:: default

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








.. GENERATED FROM PYTHON SOURCE LINES 119-123

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 125-130

.. code-block:: default

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








.. GENERATED FROM PYTHON SOURCE LINES 131-133

.. code-block:: default

    result.getIterationNumber()





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

 .. code-block:: none


    10



.. GENERATED FROM PYTHON SOURCE LINES 134-136

.. code-block:: default

    result.getOptimalPoint()






.. raw:: html

    <div class="output_subarea output_html rendered_html output_result">
    <p>[-0.0975796,0.839969]</p>
    </div>
    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 137-139

.. code-block:: default

    result.getOptimalValue()






.. raw:: html

    <div class="output_subarea output_html rendered_html output_result">
    <p>[3.00508]</p>
    </div>
    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 140-142

.. code-block:: default

    fexact






.. raw:: html

    <div class="output_subarea output_html rendered_html output_result">
    <p>[4.44089e-16]</p>
    </div>
    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 143-149

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 151-154

.. code-block:: default

    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 155-157

.. code-block:: default

    inputHistory = result.getInputSample()








.. GENERATED FROM PYTHON SOURCE LINES 158-171

.. code-block:: default

    graph = model.draw(lowerbound, upperbound, [100] * dim)
    graph.setLegends([""])
    graph.setTitle("Ackley function. Initial : black bullet. Solution : green diamond.")
    cloud = ot.Cloud(inputSample)
    cloud.setPointStyle("bullet")
    cloud.setColor("black")
    graph.add(cloud)
    cloud = ot.Cloud(inputHistory)
    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. Initial : black bullet. Solution : green diamond.
   :srcset: /auto_numerical_methods/optimization/images/sphx_glr_plot_ego_004.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 172-176

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 178-183

.. code-block:: default

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








.. GENERATED FROM PYTHON SOURCE LINES 184-186

.. code-block:: default

    result.getOptimalPoint()






.. raw:: html

    <div class="output_subarea output_html rendered_html output_result">
    <p>[4.5981e-07,0.952166]</p>
    </div>
    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 187-188

The corrected solution is now extremely accurate.

.. GENERATED FROM PYTHON SOURCE LINES 190-193

.. code-block:: default

    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 194-198

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

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

.. GENERATED FROM PYTHON SOURCE LINES 200-202

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

.. GENERATED FROM PYTHON SOURCE LINES 204-205

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

.. GENERATED FROM PYTHON SOURCE LINES 205-207

.. code-block:: default

    bm = branin_function.BraninModel()








.. GENERATED FROM PYTHON SOURCE LINES 208-209

We load the dimension,

.. GENERATED FROM PYTHON SOURCE LINES 209-211

.. code-block:: default

    dim = bm.dim








.. GENERATED FROM PYTHON SOURCE LINES 212-213

the domain boundaries,

.. GENERATED FROM PYTHON SOURCE LINES 213-216

.. code-block:: default

    lowerbound = bm.lowerbound
    upperbound = bm.upperbound








.. GENERATED FROM PYTHON SOURCE LINES 217-218

and we load the model function and its noise :

.. GENERATED FROM PYTHON SOURCE LINES 218-221

.. code-block:: default

    objectiveFunction = bm.model
    noise = bm.noiseModel








.. GENERATED FROM PYTHON SOURCE LINES 222-223

We build a sample out of the three minima :

.. GENERATED FROM PYTHON SOURCE LINES 223-225

.. code-block:: default

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








.. GENERATED FROM PYTHON SOURCE LINES 226-227

The minimum value attained `fexact` is :

.. GENERATED FROM PYTHON SOURCE LINES 227-230

.. code-block:: default

    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 231-235

.. code-block:: default

    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 236-237

The Branin function has three local minimas.

.. GENERATED FROM PYTHON SOURCE LINES 239-241

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

.. GENERATED FROM PYTHON SOURCE LINES 243-250

.. code-block:: default

    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 251-258

.. code-block:: default

    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 259-263

.. code-block:: default

    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 264-267

.. code-block:: default

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








.. GENERATED FROM PYTHON SOURCE LINES 268-270

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

.. GENERATED FROM PYTHON SOURCE LINES 272-273

We define the problem :

.. GENERATED FROM PYTHON SOURCE LINES 273-278

.. code-block:: default

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








.. GENERATED FROM PYTHON SOURCE LINES 279-280

We configure the algorithm, with the model noise:

.. GENERATED FROM PYTHON SOURCE LINES 280-283

.. code-block:: default

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








.. GENERATED FROM PYTHON SOURCE LINES 284-285

We run the algorithm and get the result:

.. GENERATED FROM PYTHON SOURCE LINES 285-288

.. code-block:: default

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








.. GENERATED FROM PYTHON SOURCE LINES 289-291

.. code-block:: default

    result.getIterationNumber()





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

 .. code-block:: none


    20



.. GENERATED FROM PYTHON SOURCE LINES 292-294

.. code-block:: default

    result.getOptimalPoint()






.. raw:: html

    <div class="output_subarea output_html rendered_html output_result">
    <p>[0.124442,0.803596]</p>
    </div>
    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 295-297

.. code-block:: default

    result.getOptimalValue()






.. raw:: html

    <div class="output_subarea output_html rendered_html output_result">
    <p>[-1.04662]</p>
    </div>
    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 298-300

.. code-block:: default

    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 301-303

.. code-block:: default

    inputHistory = result.getInputSample()








.. GENERATED FROM PYTHON SOURCE LINES 304-317

.. code-block:: default

    graph = objectiveFunction.draw(lowerbound, upperbound, [100] * dim)
    graph.setLegends([""])
    graph.setTitle("Branin function. Initial : black bullet. Solution : green diamond.")
    cloud = ot.Cloud(inputSample)
    cloud.setPointStyle("bullet")
    cloud.setColor("black")
    graph.add(cloud)
    cloud = ot.Cloud(inputHistory)
    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. Initial : black bullet. Solution : green diamond.
   :srcset: /auto_numerical_methods/optimization/images/sphx_glr_plot_ego_008.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 318-319

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 321-325

.. code-block:: default

    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






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

   **Total running time of the script:** ( 0 minutes  4.390 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-python

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

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

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