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Semi-infinite optimization: Structure and stability of the feasible set. (English) Zbl 0807.90113

Summary: The problem of the minimization of a function \(f: \mathbb{R}^ n\to \mathbb{R}\) under finitely many equality constraints and perhaps infinitely many inequality constraints gives rise to a structural analysis of the feasible set \(M[H,G]= \{x\in \mathbb{R}^ n\mid H(x)= 0,\;G(x,y)\geq 0,\;y\in Y\}\) with compact \(Y\subset \mathbb{R}^ r\). An extension of the well-known Mangasarian-Fromovitz constraint qualification (EMFCQ) is introduced. The main result for compact \(M[H,G]\) is the equivalence of the topological stability of the feasible set \([H,G]\) and the validity of EMFCQ. As a byproduct, we obtain under EMFCQ that the feasible set admits local linearizations and also that \(M[H,G]\) depends continuously on the pair \((H,G)\). Moreover, EMFCQ is shown to be satisfied generically.

MSC:

90C34 Semi-infinite programming
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