Convex semi-infinite programming: implicit optimality criterion based on the concept of immobile points

Abstract

The paper deals with convex Semi-In¯nite Programming (SIP) problems. A new concept of immobility order is introduced and an algorithm of determination of the immobility orders (DIO algorithm) and so called immobile points is suggested. It is shown that in the presence of the immobile points SIP problems do not satisfy the Slater condition. Given convex SIP problem, we determine all its immobile points and use them to formulate a Nonlinear Programming (NLP) problem in a special form. It is proved that optimality conditions for the (in¯nite) SIP problem can be formulated in terms of the analogous conditions for the corresponding (¯nite) NLP problem. The main result of the paper is the Implicit Optimality Criterion that permits to obtain new e±cient optimality conditions for the convex SIP problems (even not satisfying the Slater condition) using the known results of the optimality theory of NLP.