We consider a generalized semi-infinite optimization problem (GSIP) of the form (GSIP) min{f(x) | x ∈ M}, where M = {x ∈ ℝn | hi(x) = 0, i = 1, …, m, G(x,y) ⩾ 0, y ∈ 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(.) 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.
