The feasible set in generalized semi-infinite optimization (Q2759584)

This article surveys recent results on Generalized Semi-Infinite Optimization (GSIP) problems, which are of the form ``minimize \(f\)'' subject to \(x\in M\) (the feasible set), with NEWLINE\[NEWLINE M=\{ x\in \mathbb{R}^{n}\mid g( x,y) \leq 0,y\in Y( x) \} , NEWLINE\]NEWLINE where \(Y:\mathbb{R}^{n}\rightrightarrows \mathbb{R}^{m}\) is a set valued mapping locally bounded everywhere (in ordinary semi-infinite programming the index set \(Y( x) \) is constant). Section 1 reviews the main GSIP application and gives examples showing the abnormal features of \(M\) (it can be nonconvex and nonclosed even though all the involved functions are linear). The properties of \(M\) are analyzed from three different perspectives: orthogonal projections, description of \(M\) in terms of set valued mappings, and description of \(M\) by an optimal value function. The paper discusses the advantages of each approach in order to obtain local topological and geometrical properties of \(M\) (e.g., closedness conditions) as well as optimality conditions. NEWLINENEWLINENEWLINEThis paper is a useful introduction to GSIP theory.NEWLINENEWLINEFor the entire collection see [Zbl 0970.00036].