Shekel function

Short description: 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.[1]

The mathematical form of a function in [math]\displaystyle{ n }[/math] dimensions with [math]\displaystyle{ m }[/math] maxima is:

[math]\displaystyle{ f(\vec{x}) = \sum_{i = 1}^{m} \; \left( c_{i} + \sum\limits_{j = 1}^{n} (x_{j} - a_{ji})^2 \right)^{-1} }[/math]

or, similarly,

[math]\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} }[/math]

## Contents

* 1 Global minima
* 2 See also
* 3 References
* 4 Further reading

## Global minima

Numerically certified global minima and the corresponding solutions were obtained using interval methods for up to [math]\displaystyle{ n = 10 }[/math].[2]

## See also

* Test functions for optimization

## References

1. ↑ Molga, M.; Smutnicki, C. (2005). "Test functions for optimization needs. Test functions for optimization needs". Test Functions for Optimization Needs 101: 48. https://marksmannet.com/RobertMarks/Classes/ENGR5358/Papers/functions.pdf.
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.

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