# 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 in 1974 by Rastrigin as a 2-dimensional function and has been generalized by Rudolph. The generalized version was popularized by Hoffmeister & Bäck and Mühlenbein et al. Finding the minimum of this function is a fairly difficult problem due to its large search space and its large number of local minima.

On an $n$ -dimensional domain it is defined by:

$f(\mathbf {x} )=An+\sum _{i=1}^{n}\left[x_{i}^{2}-A\cos(2\pi x_{i})\right]$ where $A=10$ and $x_{i}\in [-5.12,5.12]$ . There are many extrema:

• The global minimum is at $\mathbf {x} =\mathbf {0}$ where $f(\mathbf {x} )=0$ .
• The maximum function value for $x_{i}\in [-5.12,5.12]$ is located around $x_{i}\in [\pm 4.52299366...,...,\pm 4.52299366...]$ :
Number of dimensions Maximum value at $\pm 4.52299366$ 1 40.35329019
2 80.70658039
3 121.0598706
4 161.4131608
5 201.7664509
6 242.1197412
7 282.4730314
8 322.8263216
9 363.1796117

Here are all the values at 0.5 interval listed for the 2D Rastrigin function with $x_{i}\in [-5.12,5.12]$ :

$f(x)$ $x1$ $0$ $\pm 0.5$ $\pm 1$ $\pm 1.5$ $\pm 2$ $\pm 2.5$ $\pm 3$ $\pm 3.5$ $\pm 4$ $\pm 4.5$ $\pm 5$ $\pm 5.12$ $x2$ $0$ 0 20.25 1 22.25 4 26.25 9 32.25 16 40.25 25 28.92
$\pm 0.5$ 20.25 40.5 21.25 42.5 24.25 46.5 29.25 52.5 36.25 60.5 45.25 49.17
$\pm 1$ 1 21.25 2 23.25 5 27.25 10 33.25 17 41.25 26 29.92
$\pm 1.5$ 22.25 42.5 23.25 44.5 26.25 48.5 31.25 54.5 38.25 62.5 47.25 51.17
$\pm 2$ 4 24.25 5 26.25 8 30.25 13 36.25 20 44.25 29 32.92
$\pm 2.5$ 26.25 46.5 27.25 48.5 30.25 52.5 35.25 58.5 42.25 66.5 51.25 55.17
$\pm 3$ 9 29.25 10 31.25 13 35.25 18 41.25 25 49.25 34 37.92
$\pm 3.5$ 32.25 52.5 33.25 54.5 36.25 58.5 41.25 64.5 48.25 72.5 57.25 61.17
$\pm 4$ 16 36.25 17 38.25 20 42.25 25 48.25 32 56.25 41 44.92
$\pm 4.5$ 40.25 60.5 41.25 62.5 44.25 66.5 49.25 72.5 56.25 80.5 65.25 69.17
$\pm 5$ 25 45.25 26 47.25 29 51.25 34 57.25 41 65.25 50 53.92
$\pm 5.12$ 28.92 49.17 29.92 51.17 32.92 55.17 37.92 61.17 44.92 69.17 53.92 57.85

The abundance of local minima underlines the necessity of a global optimization algorithm when needing to find the global minimum. Local optimization algorithms are likely to get stuck in a local minimum.