# Proximal analysis

• Francis Clarke
Chapter
Part of the Graduate Texts in Mathematics book series (GTM, volume 264)

## Abstract

We proceed in this chapter to develop the calculus (and the geometry) associated with the proximal subdifferential. The reader has encountered this object in Chapter , in the context of Hilbert spaces. Let $$f:{\mathbb{R}}^{ n}\to {\mathbb{R}}_{ \infty}$$ be given, and let x∈ dom f. Recall that $$\zeta\in\,{\mathbb{R}}^{ n}$$ is a proximal subgradient of f at x if for some σ=σ(x,ζ) ⩾ 0, and for some neighborhood V=V(x,ζ) of x, we have
$$f(y)-f(x)+\sigma|\,y-x\,|^{\,2}\:\geqslant\: \langle \,\zeta,\, y-x\,\rangle ~\; \forall \,y\in\, V.$$
This is referred to as the proximal subgradient inequality. The collection of all such ζ (which may be empty) is the proximal subdifferential of f at x, denoted P f(x). The cornerstone of our development of proximal calculus is a multi-directional extension of the mean value theorem. We also study the related theory of proximal normals, establish a multiplier rule in proximal terms, and explain the connection between viscosity (or Dini) subgradients and the other generalized derivatives that we have encountered.

Manifold Hull