Calculus Rules forV-Proximal Subdifferentials in Smooth Banach Spaces

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.

Primal Lower Nice Functions in Reflexive Smooth Banach Spaces

“…We proved in [5] that all the integral spaces L p (1 < p < ∞), as well as the sequence spaces l p (1 < p < ∞), the Sobolev spaces W p,n (1 < p < ∞), and the Schatten trace ideals C p (1 < p < ∞), are V-proximal trustworthy. The following results were proven in [6] for a particular class of a V-proximal trustworthy space, but their proofs are still valid for any V-proximal trustworthy space. We state them here without proofs.…”

“…The proof of the equalities ∂ LP f (x) = ∂ LF f (x) = ∂ Lπ f (x) for any l.s.c. functions was given in Theorem 5.1 in [6]. We assume that f is V-p.l.n.…”

V-Proximal Trustworthy Banach Spaces

“…Throughout the paper, B denotes the closed unit ball in X and h• , •i is the dual pairing between X and its dual X * . We define (see [1,2]) the analytic (resp., the geometric) V-proximal subdifferential of f at x as follows:…”

Iterative Schemes for Nonconvex Quasi-Variational Problems with V-Prox-Regular Data in Banach Spaces

“…For a closed nonempty set in and ∈ , the authors in [1] defined the concept of -proximal normal cone ( ; ) (called in [1] generalised proximal normal cone) by ( ; ) = ( ), where denotes the indicator function associated with , that is, ( ) = 0 if ∈ and ( ) = +∞ if ∉ . We recall, respectively, the concepts of limiting Fréchet subdifferential and limiting -proximal subdifferential (see [11]):…”

“…where is the generalised projection defined in terms of the functional instead of the norm square (introduced and studied in the convex case in [16] and for the nonconvex case we refer to the recent paper [11]). A point ∈ is called the generalised projection of a given * ∈ * provided that…”