## 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

### MSC:

 90C30 Nonlinear programming 49K99 Optimality conditions 49J52 Nonsmooth analysis
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### References:

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