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Level proximal subdifferential, variational convexity, and pointwise Lipschitz smoothness

Honglin Luo, Xianfu Wang, Ziyuan Wang, Xinmin Yang

TL;DR

This work extends convex analysis tools to nonconvex prox-bounded functions via the level proximal subdifferential $\partial_p^\lambda f$, establishing existence, single-valuedness, and integration results that connect to proximal hulls and Moreau envelopes. It characterizes variational convexity and variational strong convexity through local monotonicity and nonexpansiveness properties of proximal mappings, enabling local convergence guarantees for variationally convex optimization algorithms such as local proximal gradient and Krasnosel'skiĭ–Mann iterations. The results yield a rich local theory linking proximal and envelope functions to variational notions, with practical implications for designing and analyzing algorithms in nonconvex settings. A novel feature is the emphasis on pointwise (as opposed to global) Lipschitz smoothness, captured through level proximal subdifferentials and envelope-cocoercivity relations, which broadens the applicability of proximal-method techniques beyond convexity.

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 Lipschitz smoothness of a function. As a powerful tool, level proximal subdifferential provides deep insights into variational analysis and optimization.

Level proximal subdifferential, variational convexity, and pointwise Lipschitz smoothness

TL;DR

This work extends convex analysis tools to nonconvex prox-bounded functions via the level proximal subdifferential , establishing existence, single-valuedness, and integration results that connect to proximal hulls and Moreau envelopes. It characterizes variational convexity and variational strong convexity through local monotonicity and nonexpansiveness properties of proximal mappings, enabling local convergence guarantees for variationally convex optimization algorithms such as local proximal gradient and Krasnosel'skiĭ–Mann iterations. The results yield a rich local theory linking proximal and envelope functions to variational notions, with practical implications for designing and analyzing algorithms in nonconvex settings. A novel feature is the emphasis on pointwise (as opposed to global) Lipschitz smoothness, captured through level proximal subdifferentials and envelope-cocoercivity relations, which broadens the applicability of proximal-method techniques beyond convexity.

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 Lipschitz smoothness of a function. As a powerful tool, level proximal subdifferential provides deep insights into variational analysis and optimization.
Paper Structure (10 sections, 35 theorems, 106 equations, 2 figures)

This paper contains 10 sections, 35 theorems, 106 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and auxiliary results
  3. Level proximal subdifferential: existence and single-valuedness
  4. Existence of level proximal subdifferential
  5. Locally single-valued level proximal subdifferential
  6. Variational convexity and variational strong convexity
  7. Algorithms for variationally convex functions
  8. Locally single-valued proximal mappings
  9. Integration of level proximal subdifferentials
  10. Pointwise Lipschitz smoothness

Key Result

Corollary 2.5

Let $f:\mathbb R^n\to\overline{\mathbb R}$ be proper, lsc and prox-bounded with threshold $\lambda_f>0$. Then Consequently, if $\exists 0<\lambda_{0}<\lambda_{f}$ such that $f+\lambda_0^{-1}j$ is convex, then $(\forall 0<\lambda\leq\lambda_{0})\ P_{\lambda}f=(\operatorname{Id}+\lambda \partial f)^{-1}$.

Figures (2)

  • Figure 1: Correspondence between convex and variationally convex settings.
  • Figure 2: Proximal operators and Moreau envelope in Example \ref{['example:fail coin']} with $\lambda=1$.

Theorems & Definitions (61)

  • Corollary 2.5
  • Proposition 2.6
  • Example 2.7
  • Proposition 2.8: optimality condition
  • Corollary 2.9: local minimizer
  • Lemma 3.1
  • Theorem 3.2: level proximal subdifferentiability at a point
  • Example 3.3
  • Lemma 3.4
  • Theorem 3.5: level proximal subdifferentiable functions
  • ...and 51 more