From linear to convex systems: consistency, Farkas' Lemma and applications

Abstract

This paper analyzes inequality systems with an arbitrary number of proper lower semicontinuous convex constraint functions and a closed convex constraint subset of a locally convex topological vector space. More in detail, starting from well-known results on linear systems (with no constraint set), the paper reviews and completes previous works on the above class of convex systems, providing consistency theorems, two new versions of Farkas’ lemma, and optimality conditions in convex optimization. A new closed cone constraint qualification is proposed. Suitable counterparts of these results for cone-convex systems are also given.