Misplaced Pages

Shekel function

Article snapshot taken from Wikipedia with creative commons attribution-sharealike license. Give it a read and then ask your questions in the chat. We can research this topic together.
Function used as a performance test problem for optimization algorithms
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 {\displaystyle n} dimensions with m {\displaystyle m} maxima is:

f ( x → ) = ∑ i = 1 m ( c i + ∑ j = 1 n ( x j − a j i ) 2 ) − 1 {\displaystyle 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 ) = ∑ i = 1 m ( c i + ∑ j = 1 n ( x j − a i j ) 2 ) − 1 {\displaystyle 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 {\displaystyle n=10} .

See also

References

  1. Molga, M.; Smutnicki, C. (2005). "Test functions for optimization needs. Test functions for optimization needs" (PDF). Test Functions for Optimization Needs. 101: 48.
  2. Vanaret C. (2015) Hybridization of interval methods and evolutionary algorithms for solving difficult optimization problems. PhD thesis. Ecole Nationale de l'Aviation Civile. Institut National Polytechnique de Toulouse, France.

Further reading

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

Stub icon

This mathematical analysis–related article is a stub. You can help Misplaced Pages by expanding it.

Stub icon

This applied mathematics–related article is a stub. You can help Misplaced Pages by expanding it.

Categories: