The Radius of Metric Subregularity

TL;DR

This paper investigates the radius of metric subregularity, a measure of problem stability under perturbations, providing new bounds and expressions for different perturbation types and norms, with applications to constraint systems.

Contribution

It introduces novel formulas and bounds for the radius of metric subregularity under various perturbations and norms, enhancing understanding of problem stability in variational analysis.

Findings

Derived bounds for Lipschitz, semismooth, and smooth perturbations.

Provided expressions involving generalized derivatives.

Applied results to general constraint systems.

Abstract

There is a basic paradigm, called here the radius of well-posedness, which quantifies the "distance" from a given well-posed problem to the set of ill-posed problems of the same kind. In variational analysis, well-posedness is often understood as a regularity property, which is usually employed to measure the effect of perturbations and approximations of a problem on its solutions. In this paper we focus on evaluating the radius of the property of metric subregularity which, in contrast to its siblings, metric regularity, strong regularity and strong subregularity, exhibits a more complicated behavior under various perturbations. We consider three kinds of perturbations: by Lipschitz continuous functions, by semismooth functions, and by smooth functions, obtaining different expressions/bounds for the radius of subregularity, which involve generalized derivatives of set-valued mappings. We also obtain different expressions when using either Frobenius or Euclidean norm to measure the radius. As an application, we evaluate the radius of subregularity of a general constraint system. Examples illustrate the theoretical findings.