# Generalized semi-infinite optimization: a first order optimality condition and examples

Hubertus Th. Jongen, Jan-J. Rückmann and Oliver Stein

Abstract. We consider a generalized semi-infinite optimization problem of the form

(GSIP) Minimize $\{f(x)|\ x\in M\}$,

where

$M=\{x\in\R^n| h_i(x)=0, i=1,...,m, G(x,y)\ge 0, y\in Y(x)\}$

and all appearing functions are continuously differentiable. Furthermore, we assume that the set $Y(x)$ is compact for all $x$ under consideration and the set-valued mapping $Y(\cdot)$ 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 $M$.