Shekel function

File:Shekel 2D.jpg

A Shekel function in 2 dimensions and with 10 maxima

The Shekel function or also Shekel's foxholes is a multidimensional, multimodal, continuous, deterministic function commonly used as a test function for testing optimization techniques.

The mathematical form of a function in $n$ dimensions with $m$ maxima is:

$f(\overrightarrow{x})=\sum _{i=1}^m{(c_i+\sum _{j=1}^n(x_j-a_{ji}{)}^2)}^{-1}$

or, similarly,

$f(x_1,x_2,\mathrm{...},x_{n-1},x_n)=\sum _{i=1}^m{(c_i+\sum _{j=1}^n(x_j-a_{ij}{)}^2)}^{-1}$

Global minima

Numerically certified global minima and the corresponding solutions were obtained using interval methods for up to $n=10$.

See also

• Test functions for optimization

Further reading

Shekel, J. 1971. "Test Functions for Multimodal Search Techniques." Fifth Annual Princeton Conference on Information Science and Systems.