Duality gap in convex programming

Abstract

In this paper, we consider general convex programming problems and give a sufficient condition for the equality of the infimum of the original problem and the supremum of its ordinary dual. This condition may be seen as a continuity assumption on the constraint sets (i.e. on the sublevel sets of the constraint function) under linear perturbation. It allows us to generalize as well as treat previous results in a unified framework. Our main result is in fact based on a quite general constraint qualification result for quasiconvex programs involving a convex objective function proven in the setting of a real normed vector space.