Calculus Rules forV-Proximal Subdifferentials in Smooth Banach Spaces

Messaoud Bounkhel

doi:10.1155/2016/1917387

Abstract

In 2010, Bounkhel et al. introduced new proximal concepts (analytic proximal subdifferential, geometric proximal subdifferential, and proximal normal cone) in reflexive smooth Banach spaces. They proved, inp-uniformly convex andq-uniformly smooth Banach spaces, the density theorem for the new concepts of proximal subdifferential and various important properties for both proximal subdifferential concepts and the proximal normal cone concept. In this paper, we establish calculus rules (fuzzy sum rule and chain rule) for both proximal subdifferentials and we prove the Bishop-Phelps theorem for the proximal normal cone. The limiting concept for both proximal subdifferentials and for the proximal normal cone is defined and studied. We prove that both limiting constructions coincide with the Mordukhovich constructions under some assumptions on the space. Applications to nonconvex minimisation problems and nonconvex variational inequalities are established.

Highlights

In [1], the authors introduced the concept of geometric proximal proximal subdifferential subdifferential and the concept of analytic in reflexive Banach spaces for x ∈ dom f
Our results extend the existing results on the usual proximal subdifferential from Hilbert spaces setting to our Banach spaces setting
We summarise in the following proposition some needed results of the analytic and geometric V-proximal subdifferentials proved in [1]

Summary

Introduction

In [1], the authors introduced the concept of geometric proximal proximal subdifferential subdifferential. In order to avoid any confusion with other existing proximal subdifferential and proximal normal cones in Banach spaces (see, e.g., [3,4,5,6]) we will call ∂Aπf, ∂Gπf, and Nπ(S; ⋅) analytic V-proximal subdifferential, geometric V-proximal subdifferential, and V-proximal normal cone, respectively. These names are more natural since these concepts are strongly based on V and we use these names in all the paper. The paper is closed by an application, in which, we derive a necessary optimality condition for nonconvex minimisation problems and nonconvex variational inequalities in terms of generalised projections

Preliminaries

Fuzzy Sum Rules

Chain Rule

Limiting V-Proximal Subdifferential and Limiting V-Proximal Normal Cone

Applications