Methods for solving multi-extremal problems (global search)

Abstract

This work considers some methods for searching for the absolute (global) minimum of functions having concave minorants on bounded convex sets. Special methods for minimizing Lipschitz and concave functions are proposed. The methods are based on deeper cuts that use second-order information. The possibility of applying some cutting schemes that solve convex programming problems for the global minimization of concave functions on bounded convex sets is analyzed. A new original class of cuts, and a new method for minimizing concave functions, are presented. The work also considers a new approach for the solution of multi-extremal problems that is different from cutting methods. It is based on the approximation of convex sets by the convex envelopes over the convex hull of a finite number of points. This work comes close to the studies by Tuy [1,8].