Generalized Farkas’ lemma and gap-free duality for minimax DC optimization with polynomials and robust quadratic optimization

Abstract

Motivated by robust (non-convex) quadratic optimization over convex quadratic constraints, in this paper, we examine minimax difference of convex (dc) optimization over convex polynomial inequalities. By way of generalizing the celebrated Farkas' lemma to inequality systems involving the maximum of dc functions and convex polynomials, we show that there is no duality gap between a minimax DC polynomial program and its associated conjugate dual problem. We then obtain strong duality under a constraint qualification. Consequently, we present characterizations of robust solutions of uncertain general non-convex quadratic optimization problems with convex quadratic constraints, including uncertain trust-region problems.