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The radius of metric regularity at infinity

Tung Minh Nguyen, Tien-Son Pham

TL;DR

The paper studies the stability of metric regularity and strong metric regularity for set-valued mappings between Banach spaces in the asymptotic, or infinity, regime. It develops a radius-theory framework at infinity, using normal cones and coderivatives at infinity, to relate how large asymptotically vanishing perturbations can be before regularity is lost. A central result is a radius inequality: the infimum of the Lipschitz slope of perturbations at infinity required to destroy metric regularity satisfies $\inf_f lip(f)(\infty) \ge 1/reg F(\infty,\bar{y})$, with equality in finite dimensions; an analogous statement holds for strong metric regularity. The work extends key finite-dimensional radius theorems to the infinity setting and establishes perturbation-stability results that have potential applications in variational analysis and optimization.

Abstract

This paper, in the setting at infinity, presents some relationships between the modulus of metric regularity and the radius of (strong) metric regularity that gives a measure of the extent to which a set-valued mapping can be perturbed before (strong) metric regularity is lost. The results given here can be viewed as versions at infinity of [2, Theorem 1.5] and [3, Theorem 4.6].

The radius of metric regularity at infinity

TL;DR

The paper studies the stability of metric regularity and strong metric regularity for set-valued mappings between Banach spaces in the asymptotic, or infinity, regime. It develops a radius-theory framework at infinity, using normal cones and coderivatives at infinity, to relate how large asymptotically vanishing perturbations can be before regularity is lost. A central result is a radius inequality: the infimum of the Lipschitz slope of perturbations at infinity required to destroy metric regularity satisfies , with equality in finite dimensions; an analogous statement holds for strong metric regularity. The work extends key finite-dimensional radius theorems to the infinity setting and establishes perturbation-stability results that have potential applications in variational analysis and optimization.

Abstract

This paper, in the setting at infinity, presents some relationships between the modulus of metric regularity and the radius of (strong) metric regularity that gives a measure of the extent to which a set-valued mapping can be perturbed before (strong) metric regularity is lost. The results given here can be viewed as versions at infinity of [2, Theorem 1.5] and [3, Theorem 4.6].

Paper Structure

This paper contains 8 sections, 9 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation and definitions
  4. Normal cones and coderivatives
  5. Normal cones and coderivatives at infinity
  6. Metric regularity at infinity
  7. Strong metric regularity at infinity
  8. Conclusions

Key Result

Theorem 1

For a mapping $F \colon X \rightrightarrows Y$ and any $(\bar{x}, \bar{y}) \in \mathrm{gph} F$ at which $\mathrm{gph} F$ is locally closed, one has If $F$ is strongly metrically regular at $(\bar{x}, \bar{y}),$ then If $X$ and $Y$ are finite-dimensional, then both inequalities become equations. Moreover, the infimum is unchanged if taken with respect to linear mappings of rank $1.$

Theorems & Definitions (23)

  • Theorem : radius theorem for (strong) metric regularity
  • Definition 2.1: normal cones
  • Definition 2.2: coderivative
  • Definition 2.3: normal cone at infinity
  • Definition 2.4: coderivative at infinity
  • Proposition 2.5
  • proof
  • Definition 3.1
  • Theorem 3.2: radius theorem for metric regularity at infinity
  • Proposition 3.3: extended Lyusternik--Graves theorem at infinity
  • ...and 13 more