Generalizations of Slater's constraint qualification for infinite convex programs

In this paper we study constraint qualifications and duality results for infinite convex programs (P) = inf{f(x): g(x) – S, x C}, whereg = (g₁,g₂) andS = S₁ ×S₂,S_i are convex cones,i = 1, 2,C is a convex subset of a vector spaceX, andf andg_i are, respectively, convex andS_i-convex,i = 1, 2. In particular, we consider the special case whenS₂ is in afinite dimensional space,g₂ is affine andS₂ is polyhedral. We show that a recently introduced simple constraint qualification, and the so-called quasi relative interior constraint qualification both extend to (P), from the special case thatg = g₂ is affine andS = S₂ is polyhedral in a finite dimensional space (the so-called partially finite program). This provides generalized Slater type conditions for (P) which are much weaker than the standard Slater condition. We exhibit the relationship between these two constraint qualifications and show how to replace the affine assumption ong₂ and the finite dimensionality assumption onS₂, by a local compactness assumption. We then introduce the notion of strong quasi relative interior to get parallel results for more general infinite dimensional programs without the local compactness assumption. Our basic tool reduces to guaranteeing the closure of the sum of two closed convex cones.