GENERALIZATIONS OF SLATERS CONSTRAINT QUALIFICATION FOR INFINITE CONVEX-PROGRAMS

In this paper we study constraint qualifications and duality results for infinite convex programs (P) mu = inf{f(x): g(x) is-an-element-of -S, x is-an-element-of C}, where g = (g1, g2) and S = S1 x S2, S(i) are convex cones, i = 1, 2, C is a convex subset of a vector space X, and f and g(i) are, respectively, convex and S(i)-convex, i = 1, 2. In particular, we consider the special case when S2 is in a finite dimensional space, g2 is affine and S2 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 that g = g2 is affine and S = 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 on g2 and the finite dimensionality assumption on S2, 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.