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Approximate Subgradients and Coderivatives in $R^n$

Borwein, David and Borwein, Jonathan M. and Wang, Shawn Xianfu (1996) Approximate Subgradients and Coderivatives in $R^n$. Set-Valued Analysis, 4 (4). pp. 375-398. ISSN 0927-6947

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      Abstract

      We show that in two dimensions or higher the Mordukhovich--Ioffe approximate subgradient and Clarke subgradient may differ almost everywhere for real--valued Lipschitz functions. Uncountably many Fr\'echet differentiable vector--valued Lipschitz functions differing by more than constants can share the same Mordukhovich--Ioffe coderivatives. Moreover, the approximate Jacobian associated with the Mordukhovich--Ioffe coderivative can be nonconvex almost everywhere for Fr\'echet differentiable vector--valued Lipschitz functions. Finally we show that for vector--valued Lipschitz functions the approximate Jacobian associated with the Mordukhovich--Ioffe coderivative can be almost everywhere disconnected.

      Item Type: Article
      Additional Information: pubdom FALSE
      Uncontrolled Keywords: subgradient, coderivative, generalized Jacobian, Lipschitz function, bump function, gauge, nowhere dense set, Lebesgue measure, disconnectedness
      Subjects: 49-xx Calculus of variations and optimal control; optimization > 49Jxx Existence theories
      26-xx Real functions > 26Bxx Functions of several variables
      52-xx Convex and discrete geometry > 52Axx General convexity
      26-xx Real functions > 26Axx Functions of one variable
      Faculty: UNSPECIFIED
      Depositing User: Users 1 not found.
      Date Deposited: 24 Nov 2003
      Last Modified: 13 Jan 2015 17:21
      URI: https://docserver.carma.newcastle.edu.au/id/eprint/152

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