General Semi-Infinite Programming: Symmetric Mangasarian Fromovitz Constraint Qualification and the Closure of the Feasible Set

The feasible set M in General Semi-Infinite Programming (GSIP) need not to be closed. This fact is well known. We introduce a natural constraint qualification, called Symmetric Mangasarian Fromovitz Constraint Qualification (Sym-MFCQ). The Sym-MFCQ is a non-trivial extension of the well-known (Extended) MFCQ for the special case of Semi-Infinite Programming (SIP) and Disjunctive Programming. Under the Sym-MFCQ the closure of M has an easy and also natural description. As a consequence, we get a description of the interior and boundary of M. The Sym-MFCQ is shown to be generic and stable under C1-perturbations of the defining functions. For the latter stability the consideration of the closure of M is essential. We introduce an appropriate notion of Karush-Kuhn-Tucker (KKT) points. We show that local minimizers are KKT-points under the Sym-MFCQ