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 [2] as well as the characterizations of local minimizers of order one which were derived by Stein and Still [4]. Moreover, we obtain a short proof for Theorem 1.1 in [1].
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 [3]. 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.
[1] 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.
[2] 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.
[3] J.-J. Rückmann, O. Stein, On linear and linearized generalized semi-infinite optimization problems, Annals of Operations Research, Vol. 101 (2001), 191-208.
[4] O. Stein, G. Still, On optimality conditions for generalized semi-infinite programming problems, Journal of Optimization Theory and Applications, Vol. 104 (2000), 443-458.