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2 results

  • A Characterization of Proximal Subgradient Set-Valued Mappings

  • R. A. Poliquin
  • Journal:
    Canadian Mathematical Bulletin / Volume 36 / Issue 1 / 01 March 1993
    Published online by Cambridge University Press:
    20 November 2018, pp. 116-122
    Print publication:
    01 March 1993
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  • In this paper we tackle the problem of identifying set-valued mappings that are subgradient set-valued mappings. We show that a set-valued mapping is the proximal subgradient mapping of a lower semicontinuous function bounded below by a quadratic if and only if it satisfies a monotone selection property.

  • The Proximal Subgradient Formula in Banach Space

  • Philip D. Loewen
  • Journal:
    Canadian Mathematical Bulletin / Volume 31 / Issue 3 / 01 September 1988
    Published online by Cambridge University Press:
    20 November 2018, pp. 353-361
    Print publication:
    01 September 1988
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  • The proximal subgradient formula is a refinement due to Rockafellar of Clarke's fundamental proximal normal formula. It expresses Clarke's generalized gradient of a lower semicontinuous function in terms of analytically simpler proximal subgradients. We use the infinite-dimensional proximal normal formula recently given by Borwein and Strojwas to derive a new version of the proximal subgradient formula in a reflexive Banach space X with Frechet differentiable and locally uniformly convex norm. Our result improves on the one given by Borwein and Strojwas by referring only to the given norm on X.