Generalized Semi-Infinite Programming: the Nonsmooth Symmetric Reduction Ansatz

The feasible set M in Generalized Semi-Infinite Programming (GSIP) need not to be closed. Under the socalled Symmetric Magasarian Fromovitz Constraint Qualification (Sym-MFCQ) its closure cl(M) can be described by means infinitely many inequality constraints of maximum-type. In this paper we introduce the Nonsmooth Symmetric Reduction Ansatz (NSRA). Under NSRA we prove that the set cl(M) can locally be described as the feasible set of a socalled Disjunctive Optimization Problem defined by finitely many inequality constraints of maximum type. This also shows the appearance of re-entrant corners in cl(M). Under Sym-MFCQ all local minimizers of GSIP are KKT-points for GSIP. We show that NSRA is generic and stable at all KKT-points and that all KKT-points are nondegenerate. The concept of (nondegenerate) KKT-points as well as a corresponding GSIP-index are introduced in this paper. In particular, a nondegenerate KKT-point is a local minimizer if and only if its GSIP-index vanishes. At local minimizers NSRA coincides with the Symmetric Reduction Ansatz (SRA) as introduced in H. Guenzel, H.Th. Jongen, O. Stein, Generalized semi-infinite programming: the Symmetric Reduction Ansatz, Optimization Letters, Vol. 2, No. 3, pp. 415-424, 2008. In comparison with SRA, the main new issue in NSRA is the following. At KKT-points different from local minimizers the Lagrange polytope at the lower level generically need not to be a singleton anymore. In fact, it will be a full dimensional simplex. This fact is crucial to provide the above mentioned local reduction to a Disjunctive Optimization Problem. Finally, we establish a local cell-attachment theorem which will be basic for the development of a global critical point theory for GSIP in future work.