On optimality conditions for generalized semi-infinite programming problems

Generalized semi-infinite optimization problems (GSIP) are considered. We g eneralize the well-known optimality conditions for minimizers of order one in standard semi-infinite programming to the GSIP case. We give necessary a nd sufficient conditions for local minimizers of order one without the assu mption of local reduction. The necessary conditions are derived along the s ame lines as the first-order necessary conditions for GSIP in a recent pape r of Jongen, Ruckmann, and Stein (Ref. 1) by assuming the so-called extende d Mangasarian-Fromovitz constraint qualification. Using the ideas of a rece nt paper of Ruckmann and Shapiro, we give short proofs of necessary and suf ficient optimality conditions for minimizers of order one under the additio nal assumption of the Mangasarian-Fromovitz constraint qualification at all local minimizers of the so-called lower-level problem.