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Weakened subdifferentials and Frechet differentiability of real functions

  • Ginchev Ivan


    (Department of Economics, University of Insubria, Italy)

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    Let X be a real Banach space and f : X ! R [ {+1}. It is well known that the Clarke subdifferential @ f(x) of the function f at x 2 int dom f is a singleton if and only if f is strongly differentiable (then @ f(x) = {Dsf(x)}, where Dsf(x) is the strong subdifferential of f at x). Simple examples show that there exist Fr´echet differentiable at x functions f, for which @ f(x) is not a singleton. In such a sense the Clarke subdifferential is not an exact generalization of the differential of a differentiable function. In the present paper we propose a new subdifferential @w f(x), called the weakened subdifferential of f at x, which preserves the nice calculus rules of the Clarke subdifferential, and for X finite dimensional, is a singleton @w f(x) = {} if and only if f is Fr´echet differentiable at x, and then  = DF f(x).

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    Paper provided by Department of Economics, University of Insubria in its series Economics and Quantitative Methods with number qf0803.

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    Length: 21 pages
    Date of creation: Apr 2008
    Date of revision:
    Handle: RePEc:ins:quaeco:qf0803
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