Draft:Sphere function

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.^[1] Because of this, the sphere function is also collectively referred to as De Jong's function^[2] or De Jong's first function.

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Sphere function of two variables

Contour

On a ${\displaystyle n}$-dimensional domain it is defined by:

${\displaystyle f({\boldsymbol {x}})=\sum _{i=1}^{n}x_{i}^{2}}$

The function is typically evaluated on the domain ${\displaystyle x_{i}\in [-5.12,5.12]}$ for all ${\displaystyle 1\leq i\leq n}$.^[3]

It has a global minimum of zero at ${\displaystyle x_{i}=0}$

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.