Nonsmooth analysis on partially ordered vector spaces. I. Convex case

Convex analysis provides the tools to extend results of differential calculus to nonsmooth real valued functions. The purpose of this article is to study those extensions for convex vector valued mappings. We study their continuity properties, develop a subdifferential calculus and a duality theory, similar to the one existing for real valued functions. We conclude with some useful deconvexification results.