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Sufficient conditions for the stability of local minimum points in nonsmooth optimization. (English) Zbl 0679.90072

Given a local solution of \(\bar x\) of the optimization problem \(\min\{f(x); x\in S\}\) with \(S\subset {\mathbb{R}}^ n\) and \(f: {\mathbb{R}}^ n\to {\mathbb{R}}\), the author states some conditions under which a perturbed problem \(\min\{\tilde f(x)\); \(x\in \tilde S\}\) has a local solution within a given distance of x. These hypotheses are essentially a growth condition on f, estimates of Lipschitz constants of f and \(\tilde f,\) an estimate of the \(L^{\infty}\) norm of \(f-\tilde f\), non-emptiness of \(\tilde S\) near \(\bar x,\) and another stability property involving S and \(\tilde S.\) This last property is satisfied when S and \(\tilde S\) are defined by a finite number of l.s.c. inequalities and continuous equalities, the corresponding mappings for S and \(\tilde S\) being close in \(L^{\infty}\). Finally when there is no equality constraint and some qualification hypothesis holds it is proved that the hypothesis concerning the non-emptiness of \(\tilde S\) near x is satisfied.
Reviewer: J.Bonnans

MSC:

90C31 Sensitivity, stability, parametric optimization
49K40 Sensitivity, stability, well-posedness
49J52 Nonsmooth analysis
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References:

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