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On second-order variational analysis of variational convexity of prox-regular functions

Helmut Gfrerer

TL;DR

This paper develops second-order characterizations of variational convexity and variational strong convexity for prox-regular functions by leveraging first-order generalized derivatives of the subgradient mapping $∂f$ under $f$-attentive convergence. It removes the need for subdifferential continuity and introduces $f$-attentive derivatives and the SC-derivative framework, yielding neighborhood-based and point-based equivalences that link variational convexity to monotonicity properties, tilt stability, and strong metric regularity of truncated subgradient mappings. The authors provide explicit formulas for the exact bound of variational convexity and for tilt stability in terms of $D_f^*(∂f)$ and the $(P,W)$-basis, enabling robust stability analysis in nonconvex settings. These results offer a cohesive second-order toolkit that connects growth conditions, stability notions, and regularity properties, with potential implications for proximal methods and generalized equations in optimization.

Abstract

Variational convexity, together with ist strong counterpart, of extended-real-valued functions has been recently introduced by Rockafellar. In this paper we present second-order characterizations of these properties, i.e., conditions using first-order generalized derivatives of the subgradient mapping. Up to now, such characterizations are only known under the assumptions of prox-regularity and subdifferential continuity and in this paper we discard the latter. To this aim we slightly modify the definitions of the generalized derivatives to be compatible with the $f$-attentive convergence appearing in the definition of subgradients. We formulate our results in terms of both coderivatives and subspace containing derivatives. We also give formulas for the exact bound of variational convexity and study relations between variational strong convexity, tilt-stable local minimizers and strong metric regularity of some truncation of the subgradient mapping.

On second-order variational analysis of variational convexity of prox-regular functions

TL;DR

This paper develops second-order characterizations of variational convexity and variational strong convexity for prox-regular functions by leveraging first-order generalized derivatives of the subgradient mapping under -attentive convergence. It removes the need for subdifferential continuity and introduces -attentive derivatives and the SC-derivative framework, yielding neighborhood-based and point-based equivalences that link variational convexity to monotonicity properties, tilt stability, and strong metric regularity of truncated subgradient mappings. The authors provide explicit formulas for the exact bound of variational convexity and for tilt stability in terms of and the -basis, enabling robust stability analysis in nonconvex settings. These results offer a cohesive second-order toolkit that connects growth conditions, stability notions, and regularity properties, with potential implications for proximal methods and generalized equations in optimization.

Abstract

Variational convexity, together with ist strong counterpart, of extended-real-valued functions has been recently introduced by Rockafellar. In this paper we present second-order characterizations of these properties, i.e., conditions using first-order generalized derivatives of the subgradient mapping. Up to now, such characterizations are only known under the assumptions of prox-regularity and subdifferential continuity and in this paper we discard the latter. To this aim we slightly modify the definitions of the generalized derivatives to be compatible with the -attentive convergence appearing in the definition of subgradients. We formulate our results in terms of both coderivatives and subspace containing derivatives. We also give formulas for the exact bound of variational convexity and study relations between variational strong convexity, tilt-stable local minimizers and strong metric regularity of some truncation of the subgradient mapping.
Paper Structure (5 sections, 15 theorems, 69 equations)

This paper contains 5 sections, 15 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Preliminaries from variational analysis
  3. $f$-attentive derivatives of the subgradient mapping
  4. On neighborhood-based second-order characterizations of variational $s$-convexity
  5. Point-based second-order characterizations of variational strong convexity and tilt-stability

Key Result

Proposition 2.3

Let $S:\mathbb{R}^n\rightrightarrows\mathbb{R}^n$ be a mapping and consider $(\bar{x},\bar{y})\in \mathrm{gph}\, S$. Then the following statements are equivalent.

Theorems & Definitions (40)

  • Definition 2.1
  • Definition 2.2: cf. Ro24
  • Proposition 2.3
  • Remark 2.4
  • Definition 2.5: cf.Ro24
  • Definition 2.6
  • Theorem 2.7
  • Remark 2.8
  • Definition 2.9
  • Proposition 2.10
  • ...and 30 more