Shekel function

{{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 commonly used as a test function for testing optimization 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

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''.