The chain rule for VU-decompositions of nonsmooth functions

TL;DR

This paper develops a chain rule within VU-theory for nonsmooth functions, enabling better analysis and computation of gradients by decomposing the space into smooth and nonsmooth parts, with applications to various complex functions.

Contribution

It introduces a chain rule for VU-decompositions of composite nonsmooth functions, advancing the understanding of their structure and gradient computation.

Findings

Provides formulas for separation, smooth perturbation, and sum of functions.

Applies the theory to norm functions, max-of-quadratic functions, and LASSO regularizations.

Enhances tools for variational analysis of nonsmooth functions.

Abstract

In Variational Analysis, VU-theory provides a set of tools that is helpful for understanding and exploiting the structure of nonsmooth functions. The theory takes advantage of the fact that at any point, the space can be separated into two orthogonal subspaces: one that describes the direction of nonsmoothness of the function, and the other on which the function behaves smoothly and has a gradient. For a composite function, this work establishes a chain rule that facilitates the computation of such gradients and characterizes the smooth subspace under reasonable conditions. From the chain rule presented, formulas for the separation, smooth perturbation and sum of functions are provided. Several nonsmooth examples are explored, including norm functions, max-of-quadratic functions and LASSO-type regularizations.