On necessary optimality conditions in a class of optimization problems. (English) Zbl 0699.90082

The author considers the problem of minimizing a locally Lipschitz objective function f over the subset of \({\mathbb{R}}^ n\) defined by \(Q=\{x\in S:\) \(0\in F(x)\}\), where \(S\subseteq {\mathbb{R}}^ n\) is closed and nonempty and F is a set-valued map with closed, possible nonconvex, values in \({\mathbb{R}}^ n\). He shows that if \(\hat x\) solves this problem and certain regularity conditions hold, then there exists a vector \(y^*\) in \({\mathbb{R}}^ n\) such that \[ 0\in \partial f(\hat x)+F^*(y^*;\hat x,0)+N_ S(\hat x); \] moreover, if the graph of F is Clarke regular at \((\hat x,0)\), then \(-y^*\in N_{F(\hat x)}(0)\). Here N and \(\partial\) denote the Clarke normal cone and generalized gradient, while \(F^*\) is the adjoint multifunction defined in terms of the contingent cone C by \[ x^*\in F^*(y^*;x,y)\Leftrightarrow (x^*,-y^*)\in (C_{\text{graph }F}(x,y))^ 0. \] Two theoretical applications are given.
Reviewer: Ph.Loewen


90C30 Nonlinear programming
49K99 Optimality conditions
49J52 Nonsmooth analysis
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