Shekel function

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.

A Shekel function in 2 dimensions and with 10 maxima

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

$f({\vec {x}})=\sum _{i=1}^{m}\;\left(c_{i}+\sum \limits _{j=1}^{n}(x_{j}-a_{ji})^{2}\right)^{-1}$

or, similarly,

$f(x_{1},x_{2},...,x_{n-1},x_{n})=\sum _{i=1}^{m}\;\left(c_{i}+\sum \limits _{j=1}^{n}(x_{j}-a_{ij})^{2}\right)^{-1}$

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.