Proximal analysis

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 7, 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.