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A characterization of strict local minimizers of order one for nonsmooth static minmax problems. (English) Zbl 1121.90123
The authors consider the following problem: $$\text{min}\{f(x)\,\vert \,x\in S \}, $$ where $$ S:= \{x\in \Bbb R^n \,\vert \, g_i(x)\leq 0, \; i=1,\dots,p \}, $$ $$ f(x):=\sup_{y\in Y} \phi(x,y), $$ $\phi:\Bbb R^n\times \Bbb R^m\to \Bbb R$,  $Y$ is a nonempty subset of $\Bbb R^m$, and $g_i:\Bbb R^n\to \Bbb R$. 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 $$ f(x) \geq f(x_0) + \beta \Vert x-x_0\Vert \quad \text{for all } x\in S, \; \Vert x-x_0\Vert \leq \beta. $$ 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.

90C46Optimality conditions, duality
49J35Minimax problems (existence)
Full Text: DOI
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