Abstract

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.

Author information

Contact details are reproduced from the original publication and may be historical.

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

Keywords

  • Convex analysis
  • generalized differentiation
  • geometric approach
  • relative interior
  • normal cone
  • subdifferential
  • coderivative
  • calculus rules

Mathematics Subject Classification

49J52, 49J53, 90C31

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Suggested citation

D. V. Cuong, B. S. Mordukhovich, N. M. Nam, G. Sandine. “Revisiting Rockafellar's Theorem on Relative Interiors of Convex Graphs with Applications to Convex Generalized Differentiation.” Journal of Convex Analysis 30 (2023), No. 3, 835–850. https://doi.org/10.68381/jca30039

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Published by Heldermann Verlag, 2023. Rights now held by Banach Press LLC.