Generalized Semi-Infinite Programming: The Nonsmooth Symmetric Reduction Ansatz

• Authors: H. TH. Jongen;V. Shikhman
• Venue: SIAM Journal on Optimization
• Year: 2011

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 $\overline{M}$ 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 $\overline{M}$ 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 $\overline{M}$. 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. Günzel, H. Th. Jongen, and O. Stein, Optim. Lett., 2 (2008), pp. 415-424]. 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 the basis for the development of a global critical point theory for GSIP.