Generalized Semi-Infinite Programming:the Nonsmooth Symmetric Reduction Ansatz

摘要

The feasible set M in Generalized Semi-Infinite Programming (GSIP) need not be closed. Under the so-called Symmetric Mangasarian-Fromovitz Constraint Qualification (Sym- MFCQ) its closure M—can be described by means of 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 M— can locally be described as the feasible set of a so-called Disjunctive Optimization Problem defined by finitely many inequality constraints of maximum type. This also shows the appearance of re-entrant corners in 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 [1]. 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 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.