Convex Semi-Infinite programming: explicit optimality conditions

Abstract

We consider the convex Semi-In¯nite Programming (SIP) problem where objec- tive function and constraint function are convex w.r.t. a ¯nite-dimensional variable x and all of these functions are su±ciently smooth in their domains. The constraint function depends also on so called time variable t that is de¯ned on the compact set T ½ R. In our recent paper [15] the new concept of immobility order of the points of the set T was introduced and the Implicit Optimality Criterion was proved for the convex SIP problem under consideration. In this paper the Implicit Optimality Criterion is used to obtain new ¯rst and second order explicit optimality conditions. We consider separately problems that satisfy and that do not satisfy the the Slater condition. In the case of SIP problems with linear w.r.t. x constraints the optimal- ity conditions have a form of the criterion. Comparison of the results obtained with some other known optimality conditions for SIP problems is provided as well.