Higher-Order Metric Subregularity and Its Applications

TL;DR

This paper extends the theory of metric subregularity to higher orders for set-valued mappings, providing new characterizations, sensitivity analysis, and applications to Newton-type methods, especially for cases where the order exceeds one.

Contribution

It introduces the first results on metric subregularity of order greater than one, including characterizations and sensitivity analysis, with applications to convergence rates of Newton-type algorithms.

Findings

New characterizations for q>1 metric subregularity

Sensitivity analysis under small perturbations

Applications to convergence rates of Newton methods

Abstract

This paper is devoted to the study of metric subregularity and strong subregularity of any positive order $q$ for set-valued mappings in finite and infinite dimensions. While these notions have been studied and applied earlier for $q=1$ and---to a much lesser extent---for $q\in(0,1)$, no results are available for the case $q>1$. We derive characterizations of these notions for subgradient mappings, develop their sensitivity analysis under small perturbations, and provide applications to the convergence rate of Newton-type methods for solving generalized equations.