1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
|
.. _use-case-ackley:
The Ackley test case
====================
Introduction
------------
The Ackley test case is a real function defined in dimension :math:`d` where :math:`d` is an integer.
The Ackley function is defined by the equation:
.. math::
f(\mathbf{x}) = -a \exp\left(-b\sqrt{\frac{1}{d}\sum_{i=1}^d}x_i^2\right)-\exp\left(\frac{1}{d}\sum_{i=1}^d \cos(c x_i)\right)+a+\exp(1)
for any :math:`\mathbf{x} \in [-15,15]^d`. However, we consider the smaller interval :math:`[-4,4]^d` in this example, for visual purposes.
We use the dimension :math:`d=2` with the parameters :math:`a=20`, :math:`b=0.2`, :math:`c=2\pi`.
The solution is
.. math::
\mathbf{x}^\star=(0,0,...,0)
where
.. math::
f_{min} = f(\mathbf{x}^\star) = 0.
References
----------
* Adorio, E. P., & Diliman, U. P. MVF - Multivariate Test Functions Library in C for Unconstrained Global Optimization (2005). Retrieved June 2013, from http://www.geocities.ws/eadorio/mvf.pdf.
* 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.
* Back, T. (1996). Evolutionary algorithms in theory and practice: evolution strategies, evolutionary programming, genetic algorithms. Oxford University Press on Demand.
API documentation
-----------------
.. currentmodule:: openturns.usecases.ackley_function
.. autoclass:: AckleyModel
:noindex:
Examples based on this use case
-------------------------------
.. minigallery:: openturns.usecases.ackley_function.AckleyModel
|
