Algebraic Core and Convex Calculus without Topology

TL;DR

This paper develops an algebraic approach to convex analysis in vector spaces without topology, introducing algebraic core concepts and deriving new calculus tools for convex functions and optimization problems.

Contribution

It introduces the algebraic core for convex sets in vector spaces without topology and applies it to develop generalized differential calculus and subdifferential formulas.

Findings

Established equivalence between Hahn-Banach theorem and separation theorem in vector spaces.

Developed a geometric approach to generalized differential calculus for convex objects.

Derived a formula for subdifferentials of optimal value functions in parametric convex optimization.

Abstract

In this paper we study the concept of algebraic core for convex sets in general vector spaces without any topological structure and then present its applications to problems of convex analysis and optimization. Deriving the equivalence between the Hahn-Banach theorem and and a simple version of the separation theorem of convex sets in vector spaces allows us to develop a geometric approach to generalized differential calculus for convex sets, set-valued mappings, and extended-real-valued functions with qualification conditions formulated in terms of algebraic cores for such objects. We also obtain a precise formula for computing the subdifferential of optimal value functions associated with convex problems of parametric optimization in vector spaces. Functions of this type play a crucial role in many aspects of convex optimization and its applications.