Rastrigin function

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 Rudolph[2]. The generalized version was popularized by Hoffmeister & Bäck[3] and Mühlenbein et al.[4] Finding the minimum of this function is a fairly difficult problem due to its large search space and its large number of local minima.

Rastrigin function of two variables

In 3D

Contour

On an n-dimensional domain it is defined by:

${\displaystyle f(\mathbf {x} )=An+\sum _{i=1}^{n}\left[x_{i}^{2}-A\cos(2\pi x_{i})\right]}$

where ${\displaystyle A=10}$ and ${\displaystyle x_{i}\in [-5.12,5.12]}$. It has a global minimum at ${\displaystyle \mathbf {x} =\mathbf {0} }$ where ${\displaystyle f(\mathbf {x} )=0}$.

See also

• Test functions for optimization

Notes

1. Rastrigin, L. A. "Systems of extremal control." Mir, Moscow (1974).
2. G. Rudolph. "Globale Optimierung mit parallelen Evolutionsstrategien". Diplomarbeit. Department of Computer Science, University of Dortmund, July 1990.
3. F. Hoffmeister and T. Bäck. "Genetic Algorithms and Evolution Strategies: Similarities and Differences", pages 455–469 in: H.-P. Schwefel and R. Männer (eds.): Parallel Problem Solving from Nature, PPSN I, Proceedings, Springer, 1991.
4. H. Mühlenbein, D. Schomisch and J. Born. "The Parallel Genetic Algorithm as Function Optimizer ". Parallel Computing, 17, pages 619–632, 1991.