# Sphere function

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{{Short description|Optimization performance test}}

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In [mathematical optimization](/source/Mathematical_optimization), the '''sphere function''' is a [convex function](/source/Convex_function) used as a performance test problem for [optimization algorithms](/source/Optimization_algorithm). The sphere function was proposed by [Kenneth A. De Jong](/source/Kenneth_A_De_Jong) in 1975 as the first item of a series of computational test sets.<ref>{{cite thesis |last=De Jong |first=Kenneth Alan |date=1975 |title=An analysis of the behavior of a class of genetic adaptive systems |degree=PhD |publisher=University of Michigan}}</ref> Because of this, the sphere function is also collectively referred to as '''De Jong's function''' or '''De Jong's first function'''.<ref name = "MolgaMarcin2005">{{Citation |last1=Molga |first1=Marcin |last2=Smutnicki |first2=Czesław |date=2005 |title=Test functions for optimization needs |url=https://robertmarks.org/Classes/ENGR5358/Papers/functions.pdf |publisher=Robert Marks |page=2}}</ref>

On a <math>n</math>-dimensional domain it is defined by
<math display = "block">f((x_1, x_2, \dots, x_n))=\sum_{i=1}^{n}x_{i}^{2}.</math>

The function is typically evaluated on the domain <math>x_{i} \in [-5.12,5.12]</math> for all <math>1 \leq i \leq n</math>.<ref name = "MolgaMarcin2005" />

It has a [global minimum](/source/Global_minimum) of zero at <math>x_{i}=0.</math> It is a [separable function](/source/separable_function); that is, it can be expressed as a product of functions in one variable.<ref>{{cite journal| last1=Jamil | first1=Momin | last2=Yang | first2=Xin She | title=A literature survey of benchmark functions for global optimisation problems | journal=International Journal of Mathematical Modelling and Numerical Optimisation | date=2013 | volume=4 | issue=2 | page=150 | article-number=55204 |doi = 10.1504/IJMMNO.2013.055204 | arxiv=1308.4008 }}</ref>

The sphere function is used as a benchmark problem to measure algorithms' precision, [convergence rate](/source/Rate_of_convergence), and [robustness](/source/Robust_statistics), specifically over how well the algorithm handles the function's smooth nature. Several variants of the sphere function are also used, including the [Rastrigin function](/source/Rastrigin_function).<ref name = "MolgaMarcin2005" /><ref>{{cite journal| last1=Picheny | first1=Victor | last2=Wagner | first2=Tobias | last3=Ginsbourger | first3=David | title=A benchmark of kriging-based infill criteria for noisy optimization | journal=Structural and Multidisciplinary Optimization | date=2013 | volume=48 | issue=3 | pages=607–626 |doi = 10.1007/s00158-013-0919-4}}</ref>

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

== References ==
{{reflist}}

Category:Test functions for optimization
Category:Mathematical optimization
Category:Operations research

{{applied-math-stub}}

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Adapted from the Wikipedia article [Sphere function](https://en.wikipedia.org/wiki/Sphere_function) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Sphere_function?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.