# Proximal analysis

Chapter

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## 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 This is referred to as the

*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$$f(y)-f(x)+\sigma|\,y-x\,|^{\,2}\:\geqslant\: \langle \,\zeta,\, y-x\,\rangle ~\; \forall \,y\in\, V. $$

*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.### Keywords

Manifold Hull## Copyright information

© Springer-Verlag London 2013