.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_numerical_methods/optimization/plot_optimization_rosenbrock.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_optimization_rosenbrock.py>`
        to download the full example code

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

.. _sphx_glr_auto_numerical_methods_optimization_plot_optimization_rosenbrock.py:


Quick start guide to optimization
=================================

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In this example, we perform the optimization of the Rosenbrock test function.

Let :math:`a, b\in\mathbb{R}` be parameters. The Rosenbrock function is defined by

.. math::
   f(x_1, x_2) = (a-x_1)^2 + b(x_2 - x_1^2)^2


for any :math:`\mathbf{x}\in\mathbb{R}^2`.
This function is often used with :math:`a=1` and :math:`b=100`. In this case, the function has a single global minimum at:

.. math::
   \mathbf{x}^\star = (1, 1)^T.


This function has a nonlinear least squares structure.

*References*

* Rosenbrock, H.H. (1960). "An automatic method for finding the greatest or least value of a function". The Computer Journal. 3 (3): 175–184.

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Definition of the function
--------------------------

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.. code-block:: Python

    import openturns as ot
    import openturns.viewer as viewer
    from matplotlib import pylab as plt

    ot.Log.Show(ot.Log.NONE)








.. GENERATED FROM PYTHON SOURCE LINES 40-42

.. code-block:: Python

    rosenbrock = ot.SymbolicFunction(["x1", "x2"], ["(1-x1)^2+100*(x2-x1^2)^2"])








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.. code-block:: Python

    x0 = [-1.0, 1.0]








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.. code-block:: Python

    xexact = ot.Point([1.0, 1.0])








.. GENERATED FROM PYTHON SOURCE LINES 49-52

.. code-block:: Python

    lowerbound = [-2.0, -2.0]
    upperbound = [2.0, 2.0]








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Plot the iso-values of the objective function
---------------------------------------------

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.. code-block:: Python

    rosenbrock = ot.MemoizeFunction(rosenbrock)








.. GENERATED FROM PYTHON SOURCE LINES 60-64

.. code-block:: Python

    graph = rosenbrock.draw(lowerbound, upperbound, [100] * 2)
    graph.setTitle("Rosenbrock function")
    view = viewer.View(graph)




.. image-sg:: /auto_numerical_methods/optimization/images/sphx_glr_plot_optimization_rosenbrock_001.png
   :alt: Rosenbrock function
   :srcset: /auto_numerical_methods/optimization/images/sphx_glr_plot_optimization_rosenbrock_001.png
   :class: sphx-glr-single-img





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We see that the minimum is on the top right of the picture and the starting point is on the top left of the picture.
Since the function has a long valley following the curve :math:`x_2 - x^2=0`, the algorithm generally have to follow the bottom of the valley.

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Create and solve the optimization problem
-----------------------------------------

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.. code-block:: Python

    problem = ot.OptimizationProblem(rosenbrock)








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.. code-block:: Python

    algo = ot.Cobyla(problem)
    algo.setMaximumRelativeError(1.0e-1)  # on x
    algo.setMaximumEvaluationNumber(50000)
    algo.setStartingPoint(x0)
    algo.run()








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.. code-block:: Python

    result = algo.getResult()








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.. code-block:: Python

    xoptim = result.getOptimalPoint()
    xoptim






.. raw:: html

    <div class="output_subarea output_html rendered_html output_result">
    class=Point name=Unnamed dimension=2 values=[0.992501,0.985019]
    </div>
    <br />
    <br />

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.. code-block:: Python

    delta = xexact - xoptim
    absoluteError = delta.norm()
    absoluteError





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

 .. code-block:: none


    0.016752861417341336



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We see that the algorithm found an accurate approximation of the solution.

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.. code-block:: Python

    result.getOptimalValue()  # f(x*)






.. raw:: html

    <div class="output_subarea output_html rendered_html output_result">
    class=Point name=Unnamed dimension=1 values=[5.63863e-05]
    </div>
    <br />
    <br />

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.. code-block:: Python

    result.getEvaluationNumber()





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

 .. code-block:: none


    10520



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.. code-block:: Python

    graph = rosenbrock.draw(lowerbound, upperbound, [100] * 2)
    cloud = ot.Cloud(ot.Sample([x0, xoptim]))
    cloud.setColor("black")
    cloud.setPointStyle("bullet")
    graph.add(cloud)
    graph.setTitle("Rosenbrock function")
    view = viewer.View(graph)




.. image-sg:: /auto_numerical_methods/optimization/images/sphx_glr_plot_optimization_rosenbrock_002.png
   :alt: Rosenbrock function
   :srcset: /auto_numerical_methods/optimization/images/sphx_glr_plot_optimization_rosenbrock_002.png
   :class: sphx-glr-single-img





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We see that the algorithm had to start from the top left of the banana and go to the top right.

.. GENERATED FROM PYTHON SOURCE LINES 116-119

.. code-block:: Python

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




.. image-sg:: /auto_numerical_methods/optimization/images/sphx_glr_plot_optimization_rosenbrock_003.png
   :alt: Optimal value history
   :srcset: /auto_numerical_methods/optimization/images/sphx_glr_plot_optimization_rosenbrock_003.png
   :class: sphx-glr-single-img





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The function value history make the path of the algorithm clear.
In the first step, the algorithm went in the valley, which made the function value decrease rapidly.
Once there, the algorithm had to follow the bottom of the valley so that the function decreased but slowly.
In the final steps, the algorithm found the neighbourhood of the minimum so that the local convergence could take place.

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In order to see where the function was evaluated, we use the `getInputSample` method.

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.. code-block:: Python

    inputSample = result.getInputSample()








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.. code-block:: Python

    graph = rosenbrock.draw(lowerbound, upperbound, [100] * 2)
    graph.setTitle("Rosenbrock function solved with Cobyla")
    cloud = ot.Cloud(inputSample)
    graph.add(cloud)
    view = viewer.View(graph)




.. image-sg:: /auto_numerical_methods/optimization/images/sphx_glr_plot_optimization_rosenbrock_004.png
   :alt: Rosenbrock function solved with Cobyla
   :srcset: /auto_numerical_methods/optimization/images/sphx_glr_plot_optimization_rosenbrock_004.png
   :class: sphx-glr-single-img





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We see that the algorithm made lots of evaluations in the bottom of the valley before getting in the neighbourhood of the minimum.

.. GENERATED FROM PYTHON SOURCE LINES 142-149

Solving the problem with NLopt
------------------------------

We see that the `Cobyla` algorithm required lots of function evaluations.
This is why we now use the `NLopt` class with the LBFGS algorithm.
However, the algorithm may use input points which are far away from the input domain we used so far.
This is why we had bounds to the problem, so that the algorithm never goes to far away from the valley.

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.. code-block:: Python

    bounds = ot.Interval(lowerbound, upperbound)








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.. code-block:: Python

    problem = ot.OptimizationProblem(rosenbrock)
    problem.setBounds(bounds)








.. GENERATED FROM PYTHON SOURCE LINES 158-162

.. code-block:: Python

    algo = ot.NLopt(problem, "LD_LBFGS")
    algo.setStartingPoint(x0)
    algo.run()








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.. code-block:: Python

    result = algo.getResult()








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.. code-block:: Python

    xoptim = result.getOptimalPoint()
    xoptim






.. raw:: html

    <div class="output_subarea output_html rendered_html output_result">
    class=Point name=Unnamed dimension=2 values=[1,1]
    </div>
    <br />
    <br />

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.. code-block:: Python

    delta = xexact - xoptim
    absoluteError = delta.norm()
    absoluteError





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

 .. code-block:: none


    7.740583643426769e-12



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We see that the algorithm found an extremely accurate approximation of the solution.

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.. code-block:: Python

    result.getOptimalValue()  # f(x*)






.. raw:: html

    <div class="output_subarea output_html rendered_html output_result">
    class=Point name=Unnamed dimension=1 values=[1.77616e-23]
    </div>
    <br />
    <br />

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.. code-block:: Python

    result.getEvaluationNumber()





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

 .. code-block:: none


    44



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This number of iterations is much less than the previous experiment.

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.. code-block:: Python

    inputSample = result.getInputSample()








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.. code-block:: Python

    graph = rosenbrock.draw(lowerbound, upperbound, [100] * 2)
    graph.setTitle("Rosenbrock function solved with NLopt/LD_LBFGS")
    cloud = ot.Cloud(inputSample)
    graph.add(cloud)
    view = viewer.View(graph)

    plt.show()



.. image-sg:: /auto_numerical_methods/optimization/images/sphx_glr_plot_optimization_rosenbrock_005.png
   :alt: Rosenbrock function solved with NLopt/LD_LBFGS
   :srcset: /auto_numerical_methods/optimization/images/sphx_glr_plot_optimization_rosenbrock_005.png
   :class: sphx-glr-single-img






.. _sphx_glr_download_auto_numerical_methods_optimization_plot_optimization_rosenbrock.py:

.. only:: html

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

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

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

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

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