On optimality conditions for generalized semi-infinite programming problems

Generalized semi-infinite optimization problems (GSIP) are considered. We generalize the well-known optimality conditions for minimizers of order one in standard semi-infinite programming to the GSIP case. We give necessary and sufficient conditions for local minimizers of order one without the assumption of local reduction. The necessary conditions are derived along the same lines as the first-order necessary conditions for GSIP in a recent paper of Jongen, Ruckmann, and Stein (Ref. 1) by assuming the so-called extended Mangasarian-Fromovitz constraint qualification. Using the ideas of a recent paper of Ruckmann and Shapiro, we give short proofs of necessary and sufficient optimality conditions for minimizers of order one under the additional assumption of the Mangasarian-Fromovitz constraint qualification at all local minimizers of the so-called lower-level problem.