First-order optimality conditions for degenerate index sets in generalized semi-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 generalize necessary conditions given by Ruckmann and Shapiro as well as the characterizations of local minimizers of order one, which were derived by Stein and Still. 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 on convex, we show how the general optimality conditions can be strengthened, thereby giving a generalization of Theorem 4.2 in Ruckmann and Stein. Finally, if the directional derivative of a certain optimal value function exists and is subadditive with respect to the direction, we propose a Mangasarian–Fromovitz-type constraint qualification and show that it implies an Abadie-type constraint qualification.