First order optimality conditions for degenerate index sets in generalized semi-infinite optimization
Abstract. 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 Rückmann 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 .
For the special case when the so-called lower level problem is convex, we show how the general optimality conditions can be strengthened, thereby giving a generalization of Theorem 4.2 in . Finally, if the directional derivative of a certain optimal value function exists and is sub-additive with respect to the direction, we propose a Mangasarian-Fromovitz-type constraint qualification and show that it implies an Abadie-type constraint qualification.
 H.Th. Jongen, J.-J. Rückmann, O. Stein, Generalized semi-infinite optimization: a first order optimality condition and examples, Mathematical Programming, Vol. 83 (1998), 145-158.
 J.-J. Rückmann, A. Shapiro, First-order optimality conditions in generalized semi-infinite programming, Journal of Optimization Theory and Applications, Vol. 101 (1999), 677-691.
 J.-J. Rückmann, O. Stein, On linear and linearized generalized semi-infinite optimization problems, Annals of Operations Research, Vol. 101 (2001), 191-208.
 O. Stein, G. Still, On optimality conditions for generalized semi-infinite programming problems, Journal of Optimization Theory and Applications, Vol. 104 (2000), 443-458.