Convex geometry, Mathematical optimization, Convex functions, Subdifferentials -- Geometrical aspects
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 for set-valued mappings and nonsmooth functions in finite dimensions. Using this important theorem by Rockafellar allows us to improve some results on generalized differentiation of set-valued mappings in  by replacing the relative interior qualifications on graphs with qualifications on domains and/or ranges.
Copyright (c) 2024 The Authors
Locate the Document
Published as: Van Cuong, D., Mordukhovich, B., Nam, N. M., & Sandine, G. (2022). Revisiting Rockafellar's Theorem on Relative Interiors of Convex Graphs with Applications to Convex Generalized Differentiation. arXiv preprint arXiv:2201.10689.