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

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 of B. S. Mordukhovich and N. M. Nam [Geometric approach to convex subdifferential calculus, Optimization 66 (2017) 839–873] by replacing the relative interior qualifications on graphs with qualifications on domains and/or ranges.