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Generalized semi-infinite optimization: A first order optimality condition and examples. (English) Zbl 0949.90090

Summary: We consider a Generalized Semi-Infinite Optimization Problem (GSIP) of the form

$min\left\{f\left(x\right)\mid x\in M\right\}\phantom{\rule{2.em}{0ex}}\left(\mathrm{GSIP}\right)$

where $M=\left\{x\in {ℝ}^{n}\mid {h}_{i}\left(x\right)=0$, $i=1,\cdots ,m$, $G\left(x,y\right)\ge 0$, $y\in Y\left(x\right)\right\}$ and all appearing functions are continuously differentiable. Furthermore, we assume that the set $Y\left(x\right)$ is compact for all $x$ under consideration and the set-valued mapping $Y\left(·\right)$ is upper semi-continuous. The difference with a standard semi-infinite problem lies in the $x$-dependence of the index set $Y$. We prove a first order necessary optimality condition of Fritz John type without assuming a constraint qualification or any kind of reduction approach. Moreover, we discuss some geometrical properties of the feasible set $M$.

##### MSC:
 90C34 Semi-infinite programming
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