# Shekel function

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{{short description|Function used as a performance test problem for optimization algorithms}}
right|thumb|400px|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](/source/function_(mathematics)) commonly used as a test function for testing [optimization](/source/optimization_(mathematics)) techniques.<ref>{{cite journal|last1=Molga |first1=M. |last2=Smutnicki |first2=C. |title=Test functions for optimization needs. Test functions for optimization needs |journal=Test Functions for Optimization Needs|url=https://marksmannet.com/RobertMarks/Classes/ENGR5358/Papers/functions.pdf |date=2005 |volume=101 |page=48}}</ref>

The mathematical form of a function in <math>n</math> dimensions with <math>m</math> maxima is:

<math>
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>
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>
== Global minima ==
Numerically certified global minima and the corresponding solutions were obtained using interval methods for up to <math>n = 10</math>.<ref name="vanaret2015hybridation">Vanaret C. (2015) [https://www.researchgate.net/publication/337947149_Hybridization_of_interval_methods_and_evolutionary_algorithms_for_solving_difficult_optimization_problems 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.</ref>

== See also ==
*[Test functions for optimization](/source/Test_functions_for_optimization)

Category:Test functions for optimization
Category:Functions and mappings

== References ==
<references/>

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

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