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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:

$\text{min}\left\{f\left(x\right)\phantom{\rule{0.166667em}{0ex}}|\phantom{\rule{0.166667em}{0ex}}x\in S\right\},$

where

$S:=\left\{x\in {ℝ}^{n}\phantom{\rule{0.166667em}{0ex}}|\phantom{\rule{0.166667em}{0ex}}{g}_{i}\left(x\right)\le 0,\phantom{\rule{0.277778em}{0ex}}i=1,\cdots ,p\right\},$
$f\left(x\right):=\underset{y\in Y}{sup}\phi \left(x,y\right),$

$\phi :{ℝ}^{n}×{ℝ}^{m}\to ℝ$$Y$ is a nonempty subset of ${ℝ}^{m}$, and ${g}_{i}:{ℝ}^{n}\to ℝ$.

A point ${x}_{0}\in S$ is said to be a strict local minimizer of order 1 if there exist $ϵ>0$ and $\beta >0$ such that

$f\left(x\right)\ge f\left({x}_{0}\right)+\beta \parallel x-{x}_{0}\parallel \phantom{\rule{1.em}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{all}\phantom{\rule{4.pt}{0ex}}x\in S,\phantom{\rule{0.277778em}{0ex}}\parallel x-{x}_{0}\parallel \le \beta ·$

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, duality 49J35 Minimax problems (existence)