First-order optimality conditions for degenerate index sets in generalizedsemi-infinite optimization

We present a general framework for the derivation of first-order optimality conditions in generalized semi-infinite programming. Since in our approach no constraint qualifications are assumed for the index set, we can general ize necessary conditions given by Ruckmann and Shapiro (1999) as well as th e characterizations of local minimizers of order one, which were derived by Stein and Still (2000). Moreover, we obtain a short proof for Theorem 1.1 in Jongen et al. (1998). For the special case when the so-called lower-level problem is convex, we s how how the general optimality conditions can be strengthened, thereby givi ng a generalization of Theorem 4.2 in Ruckmann and Stein (2001). Finally, i f the directional derivative of a certain optimal value function exists and is subadditive with respect to the direction, we propose a Mangasarian-Fro movitz-type constraint qualification and show that it implies an Abadie-typ e constraint qualification.