On Convex Lower Level Problems in Generalized Semi-Infinite Optimization

Abstract

We give an introduction to the derivation of topological and first order properties for generalized semi-infinite optimization problems. We focus our attention on the case of convex lower level problems where the main ideas of the proofs can be illuminated without the technicalities that are necessary for the treatment of the general non-convex case, and where we can obtain stronger results.

After the description of the local topology of the feasible set around a feasible boundary point we derive approximations for the first order tangent cones to the feasible set and formulate appropriate constraint qualifications of Mangasarian-Fromovitz and Abadie type. Based upon these results we show that the first order optimality conditions given by Rückmann and Stein ([23]) for the case of linear lower level problems also hold in the jointly convex case. Moreover we prove that the set of lower level Kuhn-Tucker multipliers corresponding to a local minimizer has to be a singleton when the defining functions are in general position.