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
n
f(x)=An+
n | |
\sum | |
i=1 |
2 | |
\left[x | |
i |
-A\cos(2\pixi)\right]
A=10
xi\in[-5.12,5.12]
x=0
f(x)=0
xi\in[-5.12,5.12]
xi\in[\pm4.52299366...,...,\pm4.52299366...]
Number of dimensions | Maximum value at \pm4.52299366 | |
---|---|---|
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
xi\in[-5.12,5.12]
f(x) | x1 | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
0 | \pm0.5 | \pm1 | \pm1.5 | \pm2 | \pm2.5 | \pm3 | \pm3.5 | \pm4 | \pm4.5 | \pm5 | \pm5.12 | ||
x2 | 0 | 0 | 20.25 | 1 | 22.25 | 4 | 26.25 | 9 | 32.25 | 16 | 40.25 | 25 | 28.92 |
\pm0.5 | 20.25 | 40.5 | 21.25 | 42.5 | 24.25 | 46.5 | 29.25 | 52.5 | 36.25 | 60.5 | 45.25 | 49.17 | |
\pm1 | 1 | 21.25 | 2 | 23.25 | 5 | 27.25 | 10 | 33.25 | 17 | 41.25 | 26 | 29.92 | |
\pm1.5 | 22.25 | 42.5 | 23.25 | 44.5 | 26.25 | 48.5 | 31.25 | 54.5 | 38.25 | 62.5 | 47.25 | 51.17 | |
\pm2 | 4 | 24.25 | 5 | 26.25 | 8 | 30.25 | 13 | 36.25 | 20 | 44.25 | 29 | 32.92 | |
\pm2.5 | 26.25 | 46.5 | 27.25 | 48.5 | 30.25 | 52.5 | 35.25 | 58.5 | 42.25 | 66.5 | 51.25 | 55.17 | |
\pm3 | 9 | 29.25 | 10 | 31.25 | 13 | 35.25 | 18 | 41.25 | 25 | 49.25 | 34 | 37.92 | |
\pm3.5 | 32.25 | 52.5 | 33.25 | 54.5 | 36.25 | 58.5 | 41.25 | 64.5 | 48.25 | 72.5 | 57.25 | 61.17 | |
\pm4 | 16 | 36.25 | 17 | 38.25 | 20 | 42.25 | 25 | 48.25 | 32 | 56.25 | 41 | 44.92 | |
\pm4.5 | 40.25 | 60.5 | 41.25 | 62.5 | 44.25 | 66.5 | 49.25 | 72.5 | 56.25 | 80.5 | 65.25 | 69.17 | |
\pm5 | 25 | 45.25 | 26 | 47.25 | 29 | 51.25 | 34 | 57.25 | 41 | 65.25 | 50 | 53.92 | |
\pm5.12 | 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.