# Rastrigin function

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{{short description|Function used as a performance test problem for optimization algorithms}}
{{multiple image
   | direction = vertical
   | width     = 300
   | header    = Rastrigin function of two variables
   | image1    = Rastrigin_function.png
   | caption1  = In 3D
   | image2    = Rastrigin-smooth-contour.svg
   | caption2  = Contour
}}

In [mathematical optimization](/source/mathematical_optimization), the '''Rastrigin function''' is a non-[convex function](/source/convex_function) used as a performance test problem for [optimization algorithm](/source/optimization_algorithm)s. It is a typical example of non-linear multimodal function. It was first proposed in 1974 by Rastrigin<ref>Rastrigin, L. A. "Systems of extremal control." Mir, Moscow (1974).</ref> as a 2-dimensional function and has been generalized by Rudolph.<ref>G. Rudolph. "Globale Optimierung mit parallelen Evolutionsstrategien". Diplomarbeit. Department of Computer Science, University of Dortmund, July 1990.</ref> The generalized version was popularized by Hoffmeister &amp; Bäck<ref>F. Hoffmeister and T. Bäck. "Genetic Algorithms and Evolution Strategies: Similarities and Differences", pages 455&ndash;469 in: H.-P. Schwefel and R. Männer (eds.): Parallel Problem Solving from Nature, [PPSN](/source/Parallel_Problem_Solving_from_Nature) I, Proceedings, Springer, 1991.</ref> and Mühlenbein et al.<ref>H. Mühlenbein, D. Schomisch and J. Born. "The Parallel Genetic Algorithm as Function Optimizer ". Parallel Computing, 17, pages 619&ndash;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](/source/local_minimum).

On an <math>n</math>-dimensional domain 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>. There are many extrema:
* The global minimum is at <math>\mathbf{x} = \mathbf{0}</math> where <math>f(\mathbf{x})=0</math>.
* The maximum function value for <math>x_i\in[-5.12,5.12] </math> is located at <math>\mathbf{x} = (\pm4.52299366..., ..., \pm4.52299366...)</math>:

{| class="wikitable" 
|-
! Number of dimensions
! Maximum value at <math>\pm4.52299366</math><br />
|- style="vertical-align:bottom;"
| 1
| 40.35329019
|- style="vertical-align:bottom;"
| 2
| 80.70658039
|- style="vertical-align:bottom;"
| 3
| 121.0598706
|- style="vertical-align:bottom;"
| 4
| 161.4131608
|- style="vertical-align:bottom;"
| 5
| 201.7664509
|- style="vertical-align:bottom;"
| 6
| 242.1197412
|- style="vertical-align:bottom;"
| 7
| 282.4730314
|- style="vertical-align:bottom;"
| 8
| 322.8263216
|- style="vertical-align:bottom;"
| 9
| 363.1796117
|}

Here are all the values at 0.5 interval listed for the 2D Rastrigin function with <math>x_i\in[-5.12,5.12] </math>:

{| class="wikitable" style="font-weight:bold; vertical-align:bottom;"
|- style="text-align:center; vertical-align:middle;"
! rowspan="2" colspan="2" | <math>f(x)</math>
! colspan="12" | <math>x_1</math>
|-
| <math>0</math>
| <math>\pm0.5</math>
| <math>\pm1</math>
| <math>\pm1.5</math>
| <math>\pm2</math>
| <math>\pm2.5</math>
| <math>\pm3</math>
| <math>\pm3.5</math>
| <math>\pm4</math>
| <math>\pm4.5</math>
| <math>\pm5</math>
| <math>\pm5.12</math>
|-
| rowspan="12" style="text-align:center; vertical-align:middle;" | <math>x_2</math>
| <math>0</math>
| style="background-color:#548235; color:#FFF;" | 0
| style="background-color:#FFE699; font-weight:normal;" | 20.25
| style="background-color:#B4C6E7;" | 1
| style="background-color:#FFE699; font-weight:normal;" | 22.25
| style="background-color:#B4C6E7;" | 4
| style="background-color:#FFE699; font-weight:normal;" | 26.25
| style="background-color:#B4C6E7;" | 9
| style="background-color:#FFE699; font-weight:normal;" | 32.25
| style="background-color:#B4C6E7;" | 16
| style="background-color:#FFE699; font-weight:normal;" | 40.25
| style="background-color:#B4C6E7;" | 25
| style="background-color:#FFE699; font-weight:normal;" | 28.92
|- style="font-weight:normal;"
| style="font-weight:bold;" | <math>\pm0.5</math>
| style="background-color:#FFE699;" | 20.25
| style="background-color:#FFBDD8;" | 40.5
| style="background-color:#FFE699;" | 21.25
| style="background-color:#FFBDD8;" | 42.5
| style="background-color:#FFE699;" | 24.25
| style="background-color:#FFBDD8;" | 46.5
| style="background-color:#FFE699;" | 29.25
| style="background-color:#FFBDD8;" | 52.5
| style="background-color:#FFE699;" | 36.25
| style="background-color:#FFBDD8;" | 60.5
| style="background-color:#FFE699;" | 45.25
| style="background-color:#FFBDD8;" | 49.17
|-
| <math>\pm1</math>
| style="background-color:#B4C6E7;" | 1
| style="background-color:#FFE699; font-weight:normal;" | 21.25
| style="background-color:#B4C6E7;" | 2
| style="background-color:#FFE699; font-weight:normal;" | 23.25
| style="background-color:#B4C6E7;" | 5
| style="background-color:#FFE699; font-weight:normal;" | 27.25
| style="background-color:#B4C6E7;" | 10
| style="background-color:#FFE699; font-weight:normal;" | 33.25
| style="background-color:#B4C6E7;" | 17
| style="background-color:#FFE699; font-weight:normal;" | 41.25
| style="background-color:#B4C6E7;" | 26
| style="background-color:#FFE699; font-weight:normal;" | 29.92
|- style="font-weight:normal;"
| style="font-weight:bold;" | <math>\pm1.5</math>
| style="background-color:#FFE699;" | 22.25
| style="background-color:#FFBDD8;" | 42.5
| style="background-color:#FFE699;" | 23.25
| style="background-color:#FFBDD8;" | 44.5
| style="background-color:#FFE699;" | 26.25
| style="background-color:#FFBDD8;" | 48.5
| style="background-color:#FFE699;" | 31.25
| style="background-color:#FFBDD8;" | 54.5
| style="background-color:#FFE699;" | 38.25
| style="background-color:#FFBDD8;" | 62.5
| style="background-color:#FFE699;" | 47.25
| style="background-color:#FFBDD8;" | 51.17
|-
| <math>\pm2</math>
| style="background-color:#B4C6E7;" | 4
| style="background-color:#FFE699; font-weight:normal;" | 24.25
| style="background-color:#B4C6E7;" | 5
| style="background-color:#FFE699; font-weight:normal;" | 26.25
| style="background-color:#B4C6E7;" | 8
| style="background-color:#FFE699; font-weight:normal;" | 30.25
| style="background-color:#B4C6E7;" | 13
| style="background-color:#FFE699; font-weight:normal;" | 36.25
| style="background-color:#B4C6E7;" | 20
| style="background-color:#FFE699; font-weight:normal;" | 44.25
| style="background-color:#B4C6E7;" | 29
| style="background-color:#FFE699; font-weight:normal;" | 32.92
|- style="font-weight:normal;"
| style="font-weight:bold;" | <math>\pm2.5</math>
| style="background-color:#FFE699;" | 26.25
| style="background-color:#FFBDD8;" | 46.5
| style="background-color:#FFE699;" | 27.25
| style="background-color:#FFBDD8;" | 48.5
| style="background-color:#FFE699;" | 30.25
| style="background-color:#FFBDD8;" | 52.5
| style="background-color:#FFE699;" | 35.25
| style="background-color:#FFBDD8;" | 58.5
| style="background-color:#FFE699;" | 42.25
| style="background-color:#FFBDD8;" | 66.5
| style="background-color:#FFE699;" | 51.25
| style="background-color:#FFBDD8;" | 55.17
|-
| <math>\pm3</math>
| style="background-color:#B4C6E7;" | 9
| style="background-color:#FFE699; font-weight:normal;" | 29.25
| style="background-color:#B4C6E7;" | 10
| style="background-color:#FFE699; font-weight:normal;" | 31.25
| style="background-color:#B4C6E7;" | 13
| style="background-color:#FFE699; font-weight:normal;" | 35.25
| style="background-color:#B4C6E7;" | 18
| style="background-color:#FFE699; font-weight:normal;" | 41.25
| style="background-color:#B4C6E7;" | 25
| style="background-color:#FFE699; font-weight:normal;" | 49.25
| style="background-color:#B4C6E7;" | 34
| style="background-color:#FFE699; font-weight:normal;" | 37.92
|- style="font-weight:normal;"
| style="font-weight:bold;" | <math>\pm3.5</math>
| style="background-color:#FFE699;" | 32.25
| style="background-color:#FFBDD8;" | 52.5
| style="background-color:#FFE699;" | 33.25
| style="background-color:#FFBDD8;" | 54.5
| style="background-color:#FFE699;" | 36.25
| style="background-color:#FFBDD8;" | 58.5
| style="background-color:#FFE699;" | 41.25
| style="background-color:#FFBDD8;" | 64.5
| style="background-color:#FFE699;" | 48.25
| style="background-color:#FFBDD8;" | 72.5
| style="background-color:#FFE699;" | 57.25
| style="background-color:#FFBDD8;" | 61.17
|-
| <math>\pm4</math>
| style="background-color:#B4C6E7;" | 16
| style="background-color:#FFE699; font-weight:normal;" | 36.25
| style="background-color:#B4C6E7;" | 17
| style="background-color:#FFE699; font-weight:normal;" | 38.25
| style="background-color:#B4C6E7;" | 20
| style="background-color:#FFE699; font-weight:normal;" | 42.25
| style="background-color:#B4C6E7;" | 25
| style="background-color:#FFE699; font-weight:normal;" | 48.25
| style="background-color:#B4C6E7;" | 32
| style="background-color:#FFE699; font-weight:normal;" | 56.25
| style="background-color:#B4C6E7;" | 41
| style="background-color:#FFE699; font-weight:normal;" | 44.92
|- style="font-weight:normal;"
| style="font-weight:bold;" | <math>\pm4.5</math>
| style="background-color:#FFE699;" | 40.25
| style="background-color:#FFBDD8;" | 60.5
| style="background-color:#FFE699;" | 41.25
| style="background-color:#FFBDD8;" | 62.5
| style="background-color:#FFE699;" | 44.25
| style="background-color:#FFBDD8;" | 66.5
| style="background-color:#FFE699;" | 49.25
| style="background-color:#FFBDD8;" | 72.5
| style="background-color:#FFE699;" | 56.25
| style="background-color:#C00000; color:#FFF;" | 80.5
| style="background-color:#FFE699;" | 65.25
| style="background-color:#FFBDD8;" | 69.17
|-
| <math>\pm5</math>
| style="background-color:#B4C6E7;" | 25
| style="background-color:#FFE699; font-weight:normal;" | 45.25
| style="background-color:#B4C6E7;" | 26
| style="background-color:#FFE699; font-weight:normal;" | 47.25
| style="background-color:#B4C6E7;" | 29
| style="background-color:#FFE699; font-weight:normal;" | 51.25
| style="background-color:#B4C6E7;" | 34
| style="background-color:#FFE699; font-weight:normal;" | 57.25
| style="background-color:#B4C6E7;" | 41
| style="background-color:#FFE699; font-weight:normal;" | 65.25
| style="background-color:#B4C6E7;" | 50
| style="background-color:#FFE699; font-weight:normal;" | 53.92
|- style="font-weight:normal;"
| style="font-weight:bold;" | <math>\pm5.12</math>
| style="background-color:#FFE699;" | 28.92
| style="background-color:#FFBDD8;" | 49.17
| style="background-color:#FFE699;" | 29.92
| style="background-color:#FFBDD8;" | 51.17
| style="background-color:#FFE699;" | 32.92
| style="background-color:#FFBDD8;" | 55.17
| style="background-color:#FFE699;" | 37.92
| style="background-color:#FFBDD8;" | 61.17
| style="background-color:#FFE699;" | 44.92
| style="background-color:#FFBDD8;" | 69.17
| style="background-color:#FFE699;" | 53.92
| style="background-color:#FFBDD8;" | 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.

== See also ==
*[Test functions for optimization](/source/Test_functions_for_optimization)

==Notes==
<references/>

Category:Test functions for optimization

---
Adapted from the Wikipedia article [Rastrigin function](https://en.wikipedia.org/wiki/Rastrigin_function) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Rastrigin_function?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.