Clarke generalized derivative

In mathematics, the Clarke generalized derivatives are types generalized of derivatives that allow for the differentiation of nonsmooth functions. The Clarke derivatives were introduced by Francis Clarke in 1975.[1]

Definitions

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For a locally Lipschitz continuous function   the Clarke generalized directional derivative of   at   in the direction   is defined as   where   denotes the limit supremum.

Then, using the above definition of  , the Clarke generalized gradient of   at   (also called the Clarke subdifferential) is given as   where   represents an inner product of vectors in   Note that the Clarke generalized gradient is set-valued—that is, at each   the function value   is a set.

More generally, given a Banach space   and a subset   the Clarke generalized directional derivative and generalized gradients are defined as above for a locally Lipschitz continuous function  

See also

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References

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  1. ^ Clarke, F. H. (1975). "Generalized gradients and applications". Transactions of the American Mathematical Society. 205: 247. doi:10.1090/S0002-9947-1975-0367131-6. ISSN 0002-9947.