Global optimization

Global optimization is a branch of applied mathematics and numerical analysis that attempts to find the global minima or maxima 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 g ( x ) {displaystyle g(x)} is obviously equivalent to the minimization of the function f ( x ) := ( − 1 ) ⋅ g ( x ) {displaystyle f(x):=(-1)cdot g(x)} . Deterministic global optimization: Global optimization is a branch of applied mathematics and numerical analysis that attempts to find the global minima or maxima 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 g ( x ) {displaystyle g(x)} is obviously equivalent to the minimization of the function f ( x ) := ( − 1 ) ⋅ g ( x ) {displaystyle f(x):=(-1)cdot g(x)} . Given a possibly nonlinear and non-convex continuous function f : Ω ⊂ R n → R {displaystyle f:Omega subset mathbb {R} ^{n} o mathbb {R} } with the global minima f ∗ {displaystyle f^{*}} and the set of all global minimizers X ∗ {displaystyle X^{*}} in Ω {displaystyle Omega } , the standard minimization problem can be given as that is, finding f ∗ {displaystyle f^{*}} and a global minimizer in X ∗ {displaystyle X^{*}} ; where Ω {displaystyle Omega } is a (not necessarily convex) compact set defined by inequalities g i ( x ) ⩾ 0 , i = 1 , … , r {displaystyle g_{i}(x)geqslant 0,i=1,ldots ,r} . Global optimization is distinguished from local optimization by its focus on finding the minima or maxima over the given set, as opposed to finding local minima or maxima. Finding an arbitrary local minima is relatively straightforward by using classical local optimization methods. Finding the global minima 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. A recent approach to the global optimization problem is via minima distribution . In this work, a relationship between any continuous function f {displaystyle f} on a compact set Ω ⊂ R n {displaystyle Omega subset mathbb {R} ^{n}} and its global minima f ∗ {displaystyle f^{*}} has been strictly established. As a typical case, it follows that