Universal Duality in Conic Convex Optimization

Abstract

Given a primal-dual pair of linear programs, it is known that if their optimal values are viewed as lying on the extended real line, then the duality gap is zero, unless both problems are infeasible, in which case the optimal values are +infinity and -infinity. In contrast, for optimization problems over nonpolyhedral convex cones, a nonzero duality gap can exist even in the case where the primal and dual problems are both feasible. For a pair of dual conic convex programs, we provide simple conditions on the onstraint matricesand cone under which the duality gap is zero for every choice of linear objective function and ight-hand-side We refer to this property as niversal duality Our conditions possess the following properties: (i) they are necessary and sufficient, in the sense that if (and only if) they do not hold, the duality gap is nonzero for some linear objective function and ight-hand-side (ii) they are metrically and topologically generic; and (iii) they can be verified by solving a single conic convex program. As a side result, we also show that the feasible sets of a primal conic convex program and its dual cannot both be bounded, unless they are both empty, and we relate this to universal duality.