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In [[mathematical optimization]], the '''Rastrigin function''' is a non-[[convex function]] used as a performance test problem for optimization algorithms. It is a typical example of non-linear multimodal function. It was first proposed by Rastrigin <ref>Rastrigin, L. A. "Systems of extremal control." (1974).</ref> as a 2-dimensional function and has been generalized by Mühlenbein et al.<ref>H. Mühlenbein, D. Schomisch and J. Born. "The Parallel Genetic Algorithm as Function Optimizer ". Parallel Computing, 17, pages 619–632, 1991.</ref> Finding the minimum of this function is a fairly difficult problem due to its large search space and its large number of local minima. | |||
It is defined by: | |||
: <math>f(\mathbf{x}) = A n + \sum_{i=1}^n \left[x_i^2 - A\cos(2 \pi x_i)\right]</math> | |||
where <math>A=10</math> and <math>x_i\in[-5.12,5.12] </math>. It has a global minimum at <math>\mathbf{x} = \mathbf{0}</math> where <math>f(\mathbf{x})=0</math>. | |||
== See also == | |||
*[[Test functions for optimization]] | |||
==Notes== | |||
<references/> | |||
[[Category:Mathematical optimization]] | |||
Revision as of 09:03, 21 March 2013
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In mathematical optimization, the Rastrigin function is a non-convex function used as a performance test problem for optimization algorithms. It is a typical example of non-linear multimodal function. It was first proposed by Rastrigin [1] as a 2-dimensional function and has been generalized by Mühlenbein et al.[2] Finding the minimum of this function is a fairly difficult problem due to its large search space and its large number of local minima.
It is defined by:
![f({\mathbf {x}})=An+\sum _{{i=1}}^{n}\left[x_{i}^{2}-A\cos(2\pi x_{i})\right]](https://wikimedia.org/api/rest_v1/media/math/render/svg/1aa1c38ee739ca9cf4582867d74d469df4676cbc)
where and . It has a global minimum at where .
![x_{i}\in [-5.12,5.12]](https://wikimedia.org/api/rest_v1/media/math/render/svg/e3560b2d6c10fff0a26c994d45374c4dc70f98e5)
