Shekel function

Shekel function 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:

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

or, similarly,

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

A Shekel function in 2 dimensions and with 10 maxima

References

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