A characterization of strict local minimizers of order one for nonsmooth static minmax problems. (English) Zbl 1121.90123

Studniarski, Marcin; Taha, Abdul Whab A.

J. Math. Anal. Appl. 259, No. 2, 368-376 (2001).

The authors consider the following problem: \[ \text{min}\{f(x)\,| \,x\in S \}, \] where \[ S:= \{x\in \mathbb R^n \,| \, g_i(x)\leq 0, \; i=1,\dots,p \}, \]
\[ f(x):=\sup_{y\in Y} \phi(x,y), \] \(\phi:\mathbb R^n\times \mathbb R^m\to \mathbb R\), \(Y\) is a nonempty subset of \(\mathbb R^m\), and \(g_i:\mathbb R^n\to \mathbb R\).
A point \(x_0\in S\) is said to be a strict local minimizer of order 1 if there exist \(\epsilon > 0\) and \(\beta > 0\) such that \[ f(x) \geq f(x_0) + \beta \| x-x_0\| \quad \text{for all } x\in S, \; \| x-x_0\| \leq \beta. \] Under weak assumptions on \(\phi\) and the \(g_i\), the authors derive a necessary optimality condition for a local minimizer. Moreover, under a certain constraint qualification, a necessary and sufficient condition for a strict local minimizer of order 1 is also established. The optimality conditions are multiplier rules involving Clarke’s generalized gradient.

Reviewer: Winfried Schirotzek (Dresden)

MSC:

90C46	Optimality conditions and duality in mathematical programming
49J35	Existence of solutions for minimax problems

Keywords:

nonsmooth static minimax problems; first-order optimality conditions; strict local minimizers

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