Linearly constrained global minimization of functions with concave minorants

Abstract

In this note, we show how a recent approach for solving linearly constrained multivariate Lipschitz optimization problems and corresponding systems of inequalities can be generalized to solve optimization problems where the objective function is only assumed to possess a concave minorant at each point. This class of functions includes not only Lipschitz functions and some generalizations, such as certain ρ-convex functions and Holder functions with exponent greater than one, but also all functions which can be expressed as differences of two convex functions (d.c. functions). Thus, in particular, a new approach is obtained for the important problem of minimizing a d.c. function over a polytope.