Revisiting Rockafellar's Theorem on Relative Interiors of Convex Graphs with Applications to Convex Generalized Differentiation

TL;DR

This paper revisits Rockafellar's theorem on the relative interior of convex graphs, simplifying proofs of calculus rules in convex generalized differentiation by replacing graph qualifications with domain/range qualifications.

Contribution

It provides a new, simplified proof of Rockafellar's theorem and extends its application to calculus rules in convex generalized differentiation.

Findings

Simplified proof of Rockafellar's theorem.

Extended calculus rules for convex set-valued mappings.

Improved qualification conditions for convex analysis.

Abstract

In this paper we revisit a theorem by Rockafellar on representing the relative interior of the graph of a convex set-valued mapping in terms of the relative interior of its domain and function values. Then we apply this theorem to provide a simple way to prove many calculus rules of generalized differentiation of set-valued mappings and nonsmooth functions in finite dimensions. These results improve upon those in [14] by replacing the relative interior qualifications on graphs with qualifications on domains and/or ranges.