Editing Global optimization (section)

{{Short description|Branch of mathematics}}
{{more footnotes needed|date=December 2013}}

'''Global optimization''' is a branch of [[operations research]], [[applied mathematics]], and [[numerical analysis]] that attempts to find the global [[maximum and minimum|minimum or maximum]] of a function or a set of functions on a given set. It is usually described as a minimization problem because the maximization of the real-valued function <math>g(x)</math> is equivalent to the minimization of the function <math>f(x):=(-1)\cdot g(x)</math>.

Given a possibly nonlinear and non-convex continuous function <math>f:\Omega\subset\mathbb{R}^n\to\mathbb{R}</math> with the global minimum <math>f^*</math> and the set of all global minimizers <math>X^*</math> in <math>\Omega</math>, the standard minimization problem can be given as 
:<math>\min_{x\in\Omega}f(x),</math>
that is, finding <math>f^*</math> and a global minimizer in <math>X^*</math>; where <math>\Omega</math> is a (not necessarily convex) compact set defined by inequalities <math>g_i(x)\geqslant0, i=1,\ldots,r</math>.

Global optimization is distinguished from local optimization by its focus on finding the minimum or maximum over the given set, as opposed to finding ''local'' minima or maxima. Finding an arbitrary local minimum is relatively straightforward by using classical ''local optimization'' methods. Finding the global minimum of a function is far more difficult: analytical methods are frequently not applicable, and the use of numerical solution strategies often leads to very hard challenges.