A selection principle with applications in convex analysis and optimization

Abstract

A convex function ϑ defined on a locally convex vector space E with values in an ordered locally convex vector space F is known to be subdifferentiable at each of its points of continuity provided that (i) F is order-complete or (ii) F has the monotonic sequence property, the positive cone F + is based, and E is an Asplund space. In the present paper we establish the subdifferentiability of ϑ in the case where neither any order-completeness assumption is imposed on F nor E has to be a differentiability space, but where every positive linear functional z ̃ ϵ F ∗ is assumed continuous for σ( F, F). This includes the case where F is a reflexive ( F)-space as well as the case where F + is weakly compactly well-based. Our method of proof makes use of a selection principle for linear set-valued mappings. Part two of our paper presents a second application of the selection method. Here we obtain an infinite dimensional version of the Kuhn-Tucker theorem for convex programming.