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8 - Nonsmooth Functions and Subgradients

Published online by Cambridge University Press:  31 March 2022

Stephen J. Wright
Affiliation:
University of Wisconsin, Madison
Benjamin Recht
Affiliation:
University of California, Berkeley
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Summary

Here, we define subgradients and subdifferentials of nonsmooth functions. These are a generalization of the concept of gradients for smooth functions, that can be used as the basis of algorithms. We relate subgradients to directional derivatives and to the normal cones associated with convex sets. We introduce composite nonsmooth functions that arise in regularized optimization formulations of data analysis problems and describe optimality conditions for minimizers of these functions. Finally, we describe proximal operators and the Moreau envelope, objects associated with nonsmooth functions that are the basis of algorithms for nonsmooth optimization described in the next chapter.

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Publisher: Cambridge University Press
Print publication year: 2022

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