V-Moreau envelope of nonconvex functions on smooth Banach spaces

Abstract

We continue the study of the properties of the $ V $-Moreau envelope and generalized $ (f, \lambda) $-projection that we started in [5]. In this paper, we study the differentiability and the regularity of the $ V $-Moreau envelope and the Hölder continuity of the generalized $ (f, \lambda) $-projection. Our results extend many existing results from the convex case to the nonconvex case and from Hilbert spaces to Banach spaces. Even on Hilbert spaces and for convex functions and sets, we derived new results.

Keywords

• $ v $-moreau envelope
• generalized $ (f
• \lambda) $-projection
• $ p $-uniformly convex banach spaces
• $ q $-uniformly smooth banach spaces
• uniformly generalized prox-regular sets
• $ v $-prox-regular functions