Strong duality for generalized monotropic programming in infinite dimensions

Abstract

We establish duality results for the generalized monotropic programming problem in separated locally convex spaces. We formulate the generalized monotropic programming (GMP) as the minimization of a (possibly infinite) sum of separable proper convex functions, restricted to a closed and convex cone. We obtain strong duality under a constraint qualification based on the closedness of the sum of the epigraphs of the conjugates of the convex functions. When the objective function is the sum of finitely many proper closed convex functions, we consider two types of constraint qualifications, both of which extend those introduced in the literature. The first constraint qualification ensures strong duality, and is equivalent to the one introduced by Boţ and Wanka. The second constraint qualification is an extension of Bertsekas’ constraint qualification and we use it to prove zero duality gap.