.. _use-case-branin:

The Branin test case
====================

Introduction
------------

The Branin function is defined in 2 dimensions based on the functions :math:`g`:

.. math::
   g(u_1, u_2) = \frac{\left(u_2-5.1\frac{u_1^2}{4\pi^2}+5\frac{u_1}{\pi}-6\right)^2+10\left(1-\frac{1}{8 \pi}\right)  \cos(u_1)+10-54.8104}{51.9496}


and :math:`t`:

.. math::
   t(x_1, x2) = (15 x_1 - 5, 15 x_2)^T.


Finally, the Branin function is the composition of the two previous functions:

.. math::
   f_{Branin}(x_1, x_2) = g \circ  t(x_1, x_2)


for any :math:`\mathbf{x} \in [0, 1]^2`.

There are three global minimas:

.. math::
   \mathbf{x}^\star=(0.123895, 0.818329),


.. math::
   \mathbf{x}^\star=(0.542773, 0.151666),


and :

.. math::
   \mathbf{x}^\star=(0.961652, 0.165000)


where the function value is:

.. math::
   f_{min} = f_{Branin}(\mathbf{x}^\star) = -0.97947643837.


We assume that the output of the Branin function is noisy, with a gaussian noise.
In other words, the objective function is:

.. math::
   f(x_1, x_2) = f_{Branin}(x_1, x_2) + \epsilon


where :math:`\epsilon` is a Gaussian random variable with null mean and standard deviation :math:`\sigma_{\epsilon} = 0.1`.

References
----------

* Dixon, L. C. W., & Szego, G. P. (1978). The global optimization problem: an introduction. Towards global optimization, 2, 1-15.

* Forrester, A., Sobester, A., & Keane, A. (2008). Engineering design via surrogate modelling: a practical guide. Wiley.

* Global Optimization Test Problems. Retrieved June 2013, from http://www-optima.amp.i.kyoto-u.ac.jp/member/student/hedar/Hedar_files/TestGO.htm.

* Molga, M., & Smutnicki, C. Test functions for optimization needs (2005). Retrieved June 2013, from http://www.zsd.ict.pwr.wroc.pl/files/docs/functions.pdf.

* Picheny, V., Wagner, T., & Ginsbourger, D. (2012). A benchmark of kriging-based infill criteria for noisy optimization.


API documentation
-----------------

.. currentmodule:: openturns.usecases.branin_function

.. autoclass:: BraninModel
    :noindex:

Examples based on this use case
-------------------------------

.. minigallery:: openturns.usecases.branin_function.BraninModel