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

  • Authors:

  • F. Guerra-Vázquez;H. Th. Jongen;V. Shikhman

  • Affiliations:

  • francisco.guerra@udlap.mx;jongen@rwth-aachen.de and shikhman@mathc.rwth-aachen.de;-

  • Venue:
  • SIAM Journal on Optimization
  • Year:
  • 2010

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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.