Generalized Farkas' lemma and gap-free duality for minimax DC optimization with polynomials and robust quadratic optimization (Q280088)

In this paper, motivated by robust (non-convex) quadratic optimization over convex quadratic constraints, a minimax difference of convex (dc) optimization over convex polynomial inequalities is examined. Using a generalization of the celebrated Farkas lemma to inequality systems involving the maximum of dc functions and convex polynomials, it is shown that there is no duality gap between a minimax DC polynomial program and its associated conjugate dual problem. Then strong duality under a constraint qualification is obtained. Consequently, characterizations of robust solutions of uncertain general non-convex quadratic optimization problems with convex quadratic constraints, including uncertain trust-region problems are presented.