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

.. only:: html

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

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

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

.. _sphx_glr_advanced_mathematical_optimization_auto_examples_plot_exercise_flat_minimum.py:


Finding a minimum in a flat neighborhood
=========================================

An excercise of finding minimum. This excercise is hard because the
function is very flat around the minimum (all its derivatives are zero).
Thus gradient information is unreliable.

The function admits a minimum in [0, 0]. The challenge is to get within
1e-7 of this minimum, starting at x0 = [1, 1].

The solution that we adopt here is to give up on using gradient or
information based on local differences, and to rely on the Powell
algorithm. With 162 function evaluations, we get to 1e-8 of the
solution.

.. GENERATED FROM PYTHON SOURCE LINES 17-37

.. code-block:: default


    import numpy as np
    import scipy as sp
    import matplotlib.pyplot as plt

    def f(x):
        return np.exp(-1/(.01*x[0]**2 + x[1]**2))

    # A well-conditionned version of f:
    def g(x):
        return f([10*x[0], x[1]])

    # The gradient of g. We won't use it here for the optimization.
    def g_prime(x):
        r = np.sqrt(x[0]**2 + x[1]**2)
        return 2/r**3*g(x)*x/r

    result = sp.optimize.minimize(g, [1, 1], method="Powell", tol=1e-10)
    x_min = result.x








.. GENERATED FROM PYTHON SOURCE LINES 38-39

Some pretty plotting

.. GENERATED FROM PYTHON SOURCE LINES 39-62

.. code-block:: default


    plt.figure(0)
    plt.clf()
    t = np.linspace(-1.1, 1.1, 100)
    plt.plot(t, f([0, t]))

    plt.figure(1)
    plt.clf()
    X, Y = np.mgrid[-1.5:1.5:100j, -1.1:1.1:100j]
    plt.imshow(f([X, Y]).T, cmap=plt.cm.gray_r, extent=[-1.5, 1.5, -1.1, 1.1],
              origin='lower')
    plt.contour(X, Y, f([X, Y]), cmap=plt.cm.gnuplot)

    # Plot the gradient
    dX, dY = g_prime([.1*X[::5, ::5], Y[::5, ::5]])
    # Adjust for our preconditioning
    dX *= .1
    plt.quiver(X[::5, ::5], Y[::5, ::5], dX, dY, color='.5')

    # Plot our solution
    plt.plot(x_min[0], x_min[1], 'r+', markersize=15)

    plt.show()



.. rst-class:: sphx-glr-horizontal


    *

      .. image-sg:: /advanced/mathematical_optimization/auto_examples/images/sphx_glr_plot_exercise_flat_minimum_001.png
         :alt: plot exercise flat minimum
         :srcset: /advanced/mathematical_optimization/auto_examples/images/sphx_glr_plot_exercise_flat_minimum_001.png
         :class: sphx-glr-multi-img

    *

      .. image-sg:: /advanced/mathematical_optimization/auto_examples/images/sphx_glr_plot_exercise_flat_minimum_002.png
         :alt: plot exercise flat minimum
         :srcset: /advanced/mathematical_optimization/auto_examples/images/sphx_glr_plot_exercise_flat_minimum_002.png
         :class: sphx-glr-multi-img






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

   **Total running time of the script:** ( 0 minutes  0.119 seconds)


.. _sphx_glr_download_advanced_mathematical_optimization_auto_examples_plot_exercise_flat_minimum.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_exercise_flat_minimum.py <plot_exercise_flat_minimum.py>`

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

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


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 .. rst-class:: sphx-glr-signature

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