Sufficient optimality conditions for convex semi-infinite programming

We consider a convex semi-infinite programming (SIP) problem whose objective and constraint functions are convex w.r.t. a finite-dimensional variable x and whose constraint function also depends on a so-called index variable that ranges over a compact set in ℝ. In our previous paper (2008), we have proved an implicit optimality criterion that is based on concepts of immobile index and immobility order. This criterion permitted us to replace the optimality conditions for a feasible solution x⁰ in the convex SIP problem by similar conditions for x⁰ in certain finite nonlinear programming problems under the assumption that the active index set is finite in the original semi-infinite problem. In the present paper, we generalize the implicit optimality criterion for the case of an infinite active index set and obtain new first- and second-order sufficient optimality conditions for convex semi-infinite problems. The comparison with some other known optimality conditions is provided.