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The differentiability of real functions on normed linear space using generalized gradients

Borwein, Jonathan M. and Fitzpatrick, Simon and Giles, John (1987) The differentiability of real functions on normed linear space using generalized gradients. Journal of Optimization Theory and Applications, 128 . pp. 512-534.

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Abstract

The modification of the Clarke generalized subdifferential due to Michel and Penot is a useful tool in determining differentiability properties for certain classes of real functions on a normed linear space. The Gâteaux differentiability of any real function can be deduced from the Gâteaux differentiability of the norm if the function has a directional derivative which attains a constant related to its generalized directional derivative. For any distance function on a space with uniformly Gâteaux differentiable norm, the Clarke and Michel-Penot generalized subdifferentials at points off the set reduce to the same object and this generates a continuity characterization for Gâteaux differentiability. However, on a Banach space with rotund dual, the Fréchet differentiability of a distance function implies that it is a convex function. A mean value theorem for the modified generalized subdifferential has implications for Gâteaux differentiability.

Item Type: Article
Subjects: UNSPECIFIED
Faculty: UNSPECIFIED
Depositing User: Mrs Naghmana Tehseen
Date Deposited: 20 Feb 2015 16:20
Last Modified: 20 Feb 2015 16:20
URI: https://docserver.carma.newcastle.edu.au/id/eprint/1598

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