Moreau–Rockafellar Theorems for Nonconvex Set-Valued Maps

  • P. H. SachEmail author


In this paper, we introduce the notion of (Benson) proper subgradient of a set-valued map and prove that, for some class of nonconvex set-valued maps, a proper subgradient of the sum of two set-valued maps can be expressed as the sum of two proper subgradients of these maps. This property is also established for weak subgradients. A result in Ref. [Lin, L.J.: J. Math. Anal. Appl. 186, 30–51 (1994)], obtained under some convexity assumption, is included as a special case of the corresponding result of this paper.


Moreau–Rockafellar theorem Benson proper subgradients Set-valued maps Generalized convexity 


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© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Institute of MathematicsHanoiVietnam

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