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The Least Singular Value Function in Variational Analysis

Mario Jelitte, Boris S. Mordukhovich

TL;DR

The paper addresses the limitations of classical metric regularity in ill-posed variational problems by introducing the least singular value function $\ell_\Xi$ as a unifying framework for mixed-order regularity. It develops primal and dual (first- and second-order) tools to characterize both standard metric regularity and newer mixed-order notions, derives constructive lower bounds for the subderivative $d\ell_\Xi(\bar{\xi})(\omega)$, and connects these bounds to coderivative-based criteria for metric 2-regularity and Gfrerer regularity. The results cover polyhedral and non-polyhedral settings, provide sufficient conditions for stability in coupled constraint and variational systems, and outline pathways for algorithmic applications and further research in higher-order regularity concepts.

Abstract

Metric regularity is among the central concepts of nonlinear and variational analysis, constrained optimization, and their numerous applications. However, metric regularity can be elusive for some important ill-posed classes of problems including polynomial equations, parametric variational systems, smooth reformulations of complementarity systems with degenerate solutions, etc. The study of stability issues for such problems can often not rely on the machinery of first-order variational analysis, and so higher-order regularity concepts have been proposed in recent years. In this paper, we investigate some notions of mixed-order regularity by using advanced tools of first-order and second-order variational analysis and generalized differentiation of both primal and dual types. Efficient characterizations of such mixed-order regularity concepts are established by employing a fresh notion of the least singular value function. The obtained conditions are applied to deriving constructive criteria for mixed-order regularity in coupled constraint and variational systems.

The Least Singular Value Function in Variational Analysis

TL;DR

The paper addresses the limitations of classical metric regularity in ill-posed variational problems by introducing the least singular value function as a unifying framework for mixed-order regularity. It develops primal and dual (first- and second-order) tools to characterize both standard metric regularity and newer mixed-order notions, derives constructive lower bounds for the subderivative , and connects these bounds to coderivative-based criteria for metric 2-regularity and Gfrerer regularity. The results cover polyhedral and non-polyhedral settings, provide sufficient conditions for stability in coupled constraint and variational systems, and outline pathways for algorithmic applications and further research in higher-order regularity concepts.

Abstract

Metric regularity is among the central concepts of nonlinear and variational analysis, constrained optimization, and their numerous applications. However, metric regularity can be elusive for some important ill-posed classes of problems including polynomial equations, parametric variational systems, smooth reformulations of complementarity systems with degenerate solutions, etc. The study of stability issues for such problems can often not rely on the machinery of first-order variational analysis, and so higher-order regularity concepts have been proposed in recent years. In this paper, we investigate some notions of mixed-order regularity by using advanced tools of first-order and second-order variational analysis and generalized differentiation of both primal and dual types. Efficient characterizations of such mixed-order regularity concepts are established by employing a fresh notion of the least singular value function. The obtained conditions are applied to deriving constructive criteria for mixed-order regularity in coupled constraint and variational systems.

Paper Structure

This paper contains 3 sections, 8 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Basic Definitions and Preliminaries
  3. Subderivatives of the LSV Function

Key Result

theorem 1

Let $\mathcal{D}$ be a nonempty subset of an open set $\mathcal{U}\subset\mathbb{R}^n\times\mathbb{R}^p$, let $\Gamma:\mathcal{D}\times\mathbb{R}^m\rightrightarrows\mathbb{R}^q$ be such that $\varGamma(\xi,\cdot)$ is positively homogeneous for all $\xi\in\mathcal{D}$, and let $\varXi:\mathcal{D}\tim where $\mathcal{A}:\mathcal{U}\rightarrow\mathbb{R}^{q\times m}$. Then $\varXi(\xi,\cdot)$ is posit

Theorems & Definitions (9)

  • definition thmcounterdefinition
  • theorem 1
  • lemma thmcounterlemma
  • theorem 2
  • corollary thmcountercorollary
  • proposition thmcounterproposition
  • theorem 3
  • proposition thmcounterproposition
  • proposition thmcounterproposition