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 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 ${\displaystyle n}$-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 at ${\displaystyle \mathbf {x} =(\pm 4.52299366...,...,\pm 4.52299366...)}$:

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

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.

• Test functions for optimization