Clarke generalized derivative

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

Definitions

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 contininuous function

See Also

References

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