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

The authors consider the following problem: $\text{min}\{f(x)\,| \,x\in S \},$ 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.

### MSC:

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

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