where g=(g1,g2) and S=S1×S2, Si are convex cones, i=1,2 C is a convex subset of a vector space X, and f and gi are, respectively, convex and Si-convex, i=1,2. In particular, the authors consider the special case when S2 is in a finite dimensional space, g2 is affine and S2 is polyhedral. They 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. The authors 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. They then introduce the notion of strong quasi relative interior to get parallel results for more general infinite dimensional programs without the local compactness assumption. The present basic tool reduces to guaranteeing the closure of the sum of two closed convex cones.
