
Journal of Convex Analysis 30 (2023), No. 3, 835850 Copyright Heldermann Verlag 2023 Revisiting Rockafellar's Theorem on Relative Interiors of Convex Graphs with Applications to Convex Generalized Differentiation Dang Van Cuong Dept. of Mathematics, Faculty of Natural Sciences, Duy Tan University, Da Nang, Vietnam dvcuong@duytan.edu.vn Boris S. Mordukhovich Dept. of Mathematics, Wayne State University, Detroit, Michigan, U.S.A. boris@math.wayne.edu Nguyen Mau Nam Fariborz Maseeh Dept. of Mathematics and Statistics, Portland State University, Portland, Oregon, U.S.A. mnn3@pdx.edu Gary Sandine Fariborz Maseeh Dept. of Mathematics and Statistics, Portland State University, Portland, Oregon, U.S.A. gsandine@pdx.edu We revisit a theorem by Rockafellar on representing the relative interior of the graph of a convex setvalued 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 setvalued mappings and nonsmooth functions in finite dimensions. Using this important theorem by Rockafellar allows us to improve some results on generalized differentiation of setvalued mappings of B. S. Mordukhovich and N. M. Nam [Geometric approach to convex subdifferential calculus, Optimization 66 (2017) 839873] by replacing the relative interior qualifications on graphs with qualifications on domains and/or ranges. Keywords: Convex analysis, generalized differentiation, geometric approach, relative interior, normal cone, subdifferential, coderivative, calculus rules. MSC: 49J52, 49J53, 90C31. [ Fulltextpdf (134 KB)] for subscribers only. 