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Partial Smoothness, Subdifferentials and Set-valued Operators

Ziqi Qin, Jingwei Liang

TL;DR

The paper introduces a subdifferential-based reformulation of partial smoothness for set-valued operators, providing a refined local nondegeneracy and identifiability theory that remains valid under degeneracy and non-vanishing error. Central to the approach is a local union construction that captures perturbations along the manifold and guarantees that the resolvent maps back to the active manifold, yielding finite identification under resolvent-regularity and nondegeneracy. It extends the classical partial smoothness framework from functions to operators, establishes calculus rules for composing and combining partly smooth maps, and demonstrates identifiability results beyond traditional nondegeneracy assumptions by introducing enlarged manifolds when needed. The theory is complemented by applications to variational inequalities and proximal stochastic methods (e.g., mini-batch SGD), and it includes estimates for the identification step, illustrating practical implications for algorithm design and analysis in nonsmooth settings.

Abstract

Over the past decades, the concept "partial smoothness" has been playing as a powerful tool in several fields involving nonsmooth analysis, such as nonsmooth optimization, inverse problems and operation research, etc. The essence of partial smoothness is that it builds an elegant connection between the optimization variable and the objective function value through the subdifferential. Identifiability is the most appealing property of partial smoothness, as locally it allows us to conduct much finer or even sharp analysis, such as linear convergence or sensitivity analysis. However, currently the identifiability relies on non-degeneracy condition and exact dual convergence, which limits the potential application of partial smoothness. In this paper, we provide an alternative characterization of partial smoothness through only subdifferentials. This new perspective enables us to establish stronger identification results, explain identification under degeneracy and non-vanishing error. Moreover, we can generalize this new characterization to set-valued operators, and provide a complement definition of partly smooth operator proposed in [14].

Partial Smoothness, Subdifferentials and Set-valued Operators

TL;DR

The paper introduces a subdifferential-based reformulation of partial smoothness for set-valued operators, providing a refined local nondegeneracy and identifiability theory that remains valid under degeneracy and non-vanishing error. Central to the approach is a local union construction that captures perturbations along the manifold and guarantees that the resolvent maps back to the active manifold, yielding finite identification under resolvent-regularity and nondegeneracy. It extends the classical partial smoothness framework from functions to operators, establishes calculus rules for composing and combining partly smooth maps, and demonstrates identifiability results beyond traditional nondegeneracy assumptions by introducing enlarged manifolds when needed. The theory is complemented by applications to variational inequalities and proximal stochastic methods (e.g., mini-batch SGD), and it includes estimates for the identification step, illustrating practical implications for algorithm design and analysis in nonsmooth settings.

Abstract

Over the past decades, the concept "partial smoothness" has been playing as a powerful tool in several fields involving nonsmooth analysis, such as nonsmooth optimization, inverse problems and operation research, etc. The essence of partial smoothness is that it builds an elegant connection between the optimization variable and the objective function value through the subdifferential. Identifiability is the most appealing property of partial smoothness, as locally it allows us to conduct much finer or even sharp analysis, such as linear convergence or sensitivity analysis. However, currently the identifiability relies on non-degeneracy condition and exact dual convergence, which limits the potential application of partial smoothness. In this paper, we provide an alternative characterization of partial smoothness through only subdifferentials. This new perspective enables us to establish stronger identification results, explain identification under degeneracy and non-vanishing error. Moreover, we can generalize this new characterization to set-valued operators, and provide a complement definition of partly smooth operator proposed in [14].
Paper Structure (13 sections, 14 theorems, 38 equations, 1 figure)

This paper contains 13 sections, 14 theorems, 38 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Contribution
  3. Notations and preliminaries
  4. Set-valued operators and motononicity
  5. Continuity of set-valued operators
  6. Monotoncity of set-valued operators
  7. Prox-regularity
  8. Partial smoothness
  9. Partly smooth functions
  10. Partly smooth operators
  11. Calculus rules
  12. Identifiability
  13. Identifiability of partly smooth set-valued operators

Key Result

lemma thmcounterlemma

Let $f:\mathbb{R}^{n} \to \bar{\mathbb{R}}$, and $x\in\mathrm{dom}(f)$ with $f(x)$ finite, then both $\partial^{F} f(x), \partial f(x$) are closed, with $\partial^{F} f(x)$ being convex and $\partial^{F} f(x) \subset \partial f(x)$. If $f$ is proper and lower semi-continuous functions, then its (lim

Figures (1)

  • Figure 1: Example of partly smooth set-valued maximally monotone operator and identifiable set.

Theorems & Definitions (35)

  • definition thmcounterdefinition: Lower semi-continuity
  • definition thmcounterdefinition: Upper semi-continuity
  • lemma thmcounterlemma: rockafellar1998variational
  • definition thmcounterdefinition
  • lemma thmcounterlemma: baillon1977quelques
  • lemma thmcounterlemma: bauschke2011convex
  • definition thmcounterdefinition: Prox-Regularity poliquin1996prox
  • lemma thmcounterlemma: poliquin1996prox
  • definition thmcounterdefinition: Localization
  • definition thmcounterdefinition: Resolvent-regularity
  • ...and 25 more