Perturbation of error bounds

TL;DR

This paper extends the understanding of error bounds in Banach spaces by analyzing their stability under various perturbations, introducing new concepts and providing characterizations and examples.

Contribution

It introduces new perturbation concepts for error bounds and characterizes their stability, extending prior work in Banach space optimization.

Findings

Error bounds are stable under certain perturbations.

New perturbation concepts help estimate the radius of error bounds.

Characterizations are supported by illustrative examples.

Abstract

Our aim in the current article is to extend the developments in Kruger, Ngai & Th\'era, SIAM J. Optim. 20(6), 3280-3296 (2010) and, more precisely, to characterize, in the Banach space setting, the stability of the local and global error bound property of inequalities determined by proper lower semicontinuous under data perturbations. We propose new concepts of (arbitrary, convex and linear) perturbations of the given function defining the system under consideration, which turn out to be a useful tool in our analysis. The characterizations of error bounds for families of perturbations can be interpreted as estimates of the `radius of error bounds'. The definitions and characterizations are illustrated by examples.