# Rastrigin function

Rastrigin function of two variables
In 3D
Contour

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[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.

On an ${\displaystyle n}$-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 ${\displaystyle x_{i}\in [-5.12,5.12]}$. There are many extrema:

• The global minimum is at ${\displaystyle \mathbf {x} =\mathbf {0} }$ where ${\displaystyle f(\mathbf {x} )=0}$.
• The maximum function value for ${\displaystyle x_{i}\in [-5.12,5.12]}$ is located around ${\displaystyle x_{i}\in [\pm 4.52299366...,...,\pm 4.52299366...]}$:
Number of dimensions Maximum value at ${\displaystyle \pm 4.52299366}$
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 ${\displaystyle x_{i}\in [-5.12,5.12]}$:

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