On optimality conditions for generalized semi-infinite programming problems

Oliver Stein and Georg Still

Abstract. Generalized semi-infinite optimization problems (GSIP) are considered. We generalize the well-known optimality conditions for minima of order one in standard semi-infinite programming to the case of GSIP. We give necessary and sufficient conditions for local minima of order one without the assumption of a 'local reduction'. The necessary conditions are derived along the same lines than the first order necessary conditions for GSIP in a recent paper of Jongen, Rückmann and Stein by assuming the so-called Extended Mangasarian-Fromovitz Constraint Qualification. Using the ideas of a recent paper of Rückmann and Shapiro we can give short proofs of necessary and sufficient optimality conditions for minimizers of order one under the additional assumption of Mangasarian-Fromovitz Constraint Qualification at all local minima of the so-called lower level problem.

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