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Generalized Twice Differentiability and Quadratic Bundles in Second-Order Variational Analysis

Pham Duy Khanh, Boris S. Mordukhovich, Vo Thanh Phat, Le Duc Viet

TL;DR

This work develops a robust primal-space framework for second-order variational analysis of nonsmooth functions by introducing generalized twice differentiability and revising quadratic bundles for prox-regular settings. It establishes a tight link between generalized second-order behavior and Moreau envelopes, showing that generalized twice differentiability of a prox-regular, prox-bounded function $f$ is equivalent to classical twice differentiability of its Moreau envelope $e_{\lambda} f$, via epi-convergence; it also proves the nonemptiness of quadratic bundles $quad f(\bar{x}|\bar{v})$ and relates them to the Hessian bundle in $\mathcal{C}^{1,1}$ cases. The paper provides structural results, including a sum rule for quadratic bundles and density properties of the generalized second-order data, thereby enabling primal-dual characterizations of variational convexity, tilt stability, and variational sufficiency with potential numerical implications. Overall, the contributions lay a solid foundation for robust second-order analysis in optimization and variational problems, with avenues for infinite-dimensional extensions.

Abstract

In this paper, we investigate the concepts of generalized twice differentiability and quadratic bundles of nonsmooth functions that have been very recently proposed by Rockafellar in the framework of second-order variational analysis. These constructions, in contrast to second-order subdifferentials, are defined in primal spaces. We develop new techniques to study generalized twice differentiability for a broad class of prox-regular functions, establish their novel characterizations. Subsequently, quadratic bundles of prox-regular functions are shown to be nonempty, which provides the ground of potential applications in variational analysis and optimization.

Generalized Twice Differentiability and Quadratic Bundles in Second-Order Variational Analysis

TL;DR

This work develops a robust primal-space framework for second-order variational analysis of nonsmooth functions by introducing generalized twice differentiability and revising quadratic bundles for prox-regular settings. It establishes a tight link between generalized second-order behavior and Moreau envelopes, showing that generalized twice differentiability of a prox-regular, prox-bounded function is equivalent to classical twice differentiability of its Moreau envelope , via epi-convergence; it also proves the nonemptiness of quadratic bundles and relates them to the Hessian bundle in cases. The paper provides structural results, including a sum rule for quadratic bundles and density properties of the generalized second-order data, thereby enabling primal-dual characterizations of variational convexity, tilt stability, and variational sufficiency with potential numerical implications. Overall, the contributions lay a solid foundation for robust second-order analysis in optimization and variational problems, with avenues for infinite-dimensional extensions.

Abstract

In this paper, we investigate the concepts of generalized twice differentiability and quadratic bundles of nonsmooth functions that have been very recently proposed by Rockafellar in the framework of second-order variational analysis. These constructions, in contrast to second-order subdifferentials, are defined in primal spaces. We develop new techniques to study generalized twice differentiability for a broad class of prox-regular functions, establish their novel characterizations. Subsequently, quadratic bundles of prox-regular functions are shown to be nonempty, which provides the ground of potential applications in variational analysis and optimization.
Paper Structure (7 sections, 24 theorems, 179 equations)

This paper contains 7 sections, 24 theorems, 179 equations.

Table of Contents

  1. Introduction
  2. Preliminaries from Variational Analysis
  3. Variational $s$-Convexity and Second-Order Subderivatives
  4. Properties of Generalized Twice Differentiable Functions
  5. Generalized Twice Differentiability via Moreau Envelopes
  6. Quadratic Bundles
  7. Conclusions and Future Research

Key Result

Proposition 2.1

Let $\{f_k\}$ be a sequence of functions on ${\rm I\!R}^n$, and let $x\in{\rm I\!R}^n$. Then we have Therefore, $\{f_k\}$ epigraphically converges to $f$ if and only if at each point $x$ it holds In particular, if $\{f_k\}$ epigraphically converges to $f$, then for all $x$ there exists a sequence $x_k \to x$ such that $f_k(x_k) \to f(x)$ as $k\to\infty$.

Theorems & Definitions (37)

  • Proposition 2.1
  • Proposition 2.2
  • Definition 3.1
  • Theorem 3.2
  • Proposition 3.3
  • Theorem 3.4
  • Example 3.5
  • Corollary 3.6
  • Definition 4.1
  • Definition 4.2
  • ...and 27 more