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Characterizations of Variational Convexity and Tilt Stability via Quadratic Bundles

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

Abstract

In this paper, we establish characterizations of variational $s$-convexity and tilt stability for prox-regular functions in the absence of subdifferential continuity via quadratic bundles, a kind of primal-dual generalized second-order derivatives recently introduced by Rockafellar. Deriving such characterizations in the effective pointbased form requires a certain revision of quadratic bundles investigated below. Our device is based on the notion of generalized twice differentiability and its novel characterization via classical twice differentiability of the associated Moreau envelopes combined with various limiting procedures for functions and sets.

Characterizations of Variational Convexity and Tilt Stability via Quadratic Bundles

Abstract

In this paper, we establish characterizations of variational -convexity and tilt stability for prox-regular functions in the absence of subdifferential continuity via quadratic bundles, a kind of primal-dual generalized second-order derivatives recently introduced by Rockafellar. Deriving such characterizations in the effective pointbased form requires a certain revision of quadratic bundles investigated below. Our device is based on the notion of generalized twice differentiability and its novel characterization via classical twice differentiability of the associated Moreau envelopes combined with various limiting procedures for functions and sets.
Paper Structure (6 sections, 17 theorems, 77 equations)

This paper contains 6 sections, 17 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Preliminaries from Variational Analysis
  3. Generalized Twice Differentiability and Quadratic Bundles
  4. Variational $s$-Convexity via Quadratic Bundles
  5. Tilt Stability of Local Minimizers
  6. Conclusions and Future Research

Key Result

Proposition 2.1

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

Theorems & Definitions (25)

  • Proposition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Corollary 2.6
  • Proposition 3.1
  • Definition 3.2
  • Definition 3.3
  • Proposition 3.4
  • ...and 15 more