Journal of Convex Analysis 30 (2023), No. 3, 835--850
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 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.
Keywords: Convex analysis, generalized differentiation, geometric approach, relative interior, normal cone, subdifferential, coderivative, calculus rules.
MSC: 49J52, 49J53, 90C31.
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