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 {\displaystyle n} -dimensional domain it is defined by:

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

where A = 10 {\displaystyle A=10} and x i ∈ [ − 5.12 , 5.12 ] {\displaystyle x_{i}\in [-5.12,5.12]} . There are many extrema:

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

Number of dimensions	Maximum value at ± 4.52299366 {\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 x i ∈ [ − 5.12 , 5.12 ] {\displaystyle x_{i}\in [-5.12,5.12]} :

f ( x ) {\displaystyle f(x)}		x 1 {\displaystyle x_{1}}
		0 {\displaystyle 0}	± 0.5 {\displaystyle \pm 0.5}	± 1 {\displaystyle \pm 1}	± 1.5 {\displaystyle \pm 1.5}	± 2 {\displaystyle \pm 2}	± 2.5 {\displaystyle \pm 2.5}	± 3 {\displaystyle \pm 3}	± 3.5 {\displaystyle \pm 3.5}	± 4 {\displaystyle \pm 4}	± 4.5 {\displaystyle \pm 4.5}	± 5 {\displaystyle \pm 5}	± 5.12 {\displaystyle \pm 5.12}
x 2 {\displaystyle x_{2}}	0 {\displaystyle 0}	0	20.25	1	22.25	4	26.25	9	32.25	16	40.25	25	28.92
	± 0.5 {\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
	± 1 {\displaystyle \pm 1}	1	21.25	2	23.25	5	27.25	10	33.25	17	41.25	26	29.92
	± 1.5 {\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
	± 2 {\displaystyle \pm 2}	4	24.25	5	26.25	8	30.25	13	36.25	20	44.25	29	32.92
	± 2.5 {\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
	± 3 {\displaystyle \pm 3}	9	29.25	10	31.25	13	35.25	18	41.25	25	49.25	34	37.92
	± 3.5 {\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
	± 4 {\displaystyle \pm 4}	16	36.25	17	38.25	20	42.25	25	48.25	32	56.25	41	44.92
	± 4.5 {\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
	± 5 {\displaystyle \pm 5}	25	45.25	26	47.25	29	51.25	34	57.25	41	65.25	50	53.92
	± 5.12 {\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.