On duality theory of convex semi-infinite programming

where is a (possibly infinite) set, f : R ! R is an extended real valued function and g : R ! R. In the above formulation, a feasible point x 2 R is supposed to satisfy the constraints gðx,!Þ 0 for all !2 , and no structural assumptions are made about the set . In some situations it is natural to require that these constraints hold for almost every (a.e.) !2 . That is, the set is equipped with a sigma algebra F and a (finite) measure on ð ,FÞ. Then it is said that a property holds for a.e. !2 if there is a set A 2 F such that ðAÞ 1⁄4 0 and the property holds for all ! 2 nA. The formulation ‘‘for a.e. !2 ’’ is relevant, for example, in stochastic programming (cf [9–11]). There exists an extensive literature on duality of convex SIPs (see, e.g., [4] and references therein), and in particular on linear SIPs (see [2] and, for a more recent survey, [3]). In this article, we discuss an approach to duality theory of both