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Generalized Directional Derivatives and Subgradients of Nonconvex Functions

Published online by Cambridge University Press:  20 November 2018

R. T. Rockafellar
R. T. Rockafellar*
Affiliation:
University of Washington, Seattle, Washington
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Studies of optimization problems and certain kinds of differential equations have led in recent years to the development of a generalized theory of differentiation quite distinct in spirit and range of application from the one based on L. Schwartz's “distributions.” This theory associates with an extended-real-valued function ƒ on a linear topological space E and a point x ∈ E certain elements of the dual space E* called subgradients or generalized gradients of ƒ at x. These form a set ∂ƒ(x) that is always convex and weak*-closed (possibly empty). The multifunction ∂ƒ: x →∂ƒ(x) is the sub differential of ƒ.

Rules that relate ∂ƒ to generalized directional derivatives of ƒ, or allow ∂ƒ to be expressed or estimated in terms of the subdifferentials of other functions (whenƒ = ƒ1 + ƒ2,ƒ = g o A, etc.), comprise the sub differential calculus.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 2 , February 1980 , pp. 257 - 280
Copyright
Copyright © Canadian Mathematical Society 1980

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