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

Abstract

The feasible set $M$ in general semi-infinite programming (GSIP) need not 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 nontrivial 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 $\overline{M}$ 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 $C^1$-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.