Level proximal subdifferential, variational convexity, and pointwise quadratic approximation

TL;DR

This paper systematically studies the level proximal subdifferential, linking it to variational convexity and convergence properties of proximal algorithms, and explores its properties and applications in variational analysis.

Contribution

It characterizes variational convexity via local properties of the proximal mapping and subdifferential, and investigates the subdifferential's properties and its role in optimization.

Findings

Variational convexity characterized by local firm nonexpansiveness.

Proximal gradient method converges to local minimizers under variational sufficiency.

Level proximal subdifferential provides insights into variational analysis and optimization.

Abstract

Level proximal subdifferential was introduced by Rockafellar recently for studying proximal mappings of possibly nonconvex functions. In this paper a systematic study of level proximal subdifferential is given. We characterize variational convexity of a function by local firm nonexpansiveness of proximal mappings or local relative monotonicity of level proximal subdifferential, and use them to study local convergence of proximal gradient method and others for variationally convex functions. Variational sufficiency guarantees that proximal gradient method converges to local minimizers rather than just critical points. We also investigate the existence, single-valuedness and integration of level proximal subdifferential, and quantify pointwise quadratic approximation (or Lipschitz smoothness) of a function. As a powerful tool, level proximal subdifferential provides deep insights into variational analysis and optimization.