Sufficient optimality conditions for convex semi-infinite programming

Abstract

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 . In our previous paper [O.I. Kostyukova, T.V. Tchemisova, and S.A. Yermalinskaya, On the algorithm of determination of immobile indices for convex SIP problems, IJAMAS Int. J. Math. Stat. 13(J08) (2008), pp. 13–33], 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 x 0 in the convex SIP problem by similar conditions for x 0 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.