{{Short description|Optimization performance test}}
<!-- {{Original research}} --> {{multiple image | align = right | direction = vertical | width = 300 | header = Sphere function of two variables <!-- TODO: Add an image of the 3D sphere function --> <!-- | image1 = Sphere function.svg | alt1 = 3D plot of the sphere function | caption1 = In 3D | image2 = Sphere contour.svg | alt2 = Contour plot of the sphere function | caption2 = Contour --> | image1 = Sphere contour.svg | alt1 = Contour plot of the sphere function | caption1 = Contour }} In mathematical optimization, the '''sphere function''' is a convex function used as a performance test problem for optimization algorithms. The sphere function was proposed by 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 of zero at <math>x_{i}=0.</math> It is a 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, and robustness, 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.<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
== References == {{reflist}}
Category:Test functions for optimization Category:Mathematical optimization Category:Operations research
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