Primal Characterizations of Stability of Error Bounds for Semi-infinite Convex Constraint Systems in Banach Spaces

TL;DR

This paper characterizes the stability of error bounds in semi-infinite convex systems within Banach spaces using primal conditions based on directional derivatives, linking stability to minimax problem solutions and perturbation effects.

Contribution

It introduces primal criteria for the stability of error bounds and Hoffman's constants under perturbations, advancing sensitivity analysis in semi-infinite convex systems.

Findings

Stability of error bounds is equivalent to minimax problems' optimal values being away from zero.

All component functions must have the same linear perturbation for stability.

Primal criteria for bounded Hoffman's constants under data perturbations are established.

Abstract

This article is devoted to the stability of error bounds (local and global) for semi-infinite convex constraint systems in Banach spaces. We provide primal characterizations of the stability of local and global error bounds when systems are subject to small perturbations. These characterizations are given in terms of the directional derivatives of the functions that enter into the definition of these systems. It is proved that the stability of error bounds is essentially equivalent to verifying that the optimal values of several minimax problems, defined in terms of the directional derivatives of the functions defining these systems, are outside of some neighborhood of zero. Moreover, such stability only requires that all component functions entering the system have the same linear perturbation. When these stability results are applied to the sensitivity analysis of Hoffman's constants for semi-infinite linear systems, primal criteria for Hoffman's constants to be uniformly bounded under perturbations of the problem data are obtained.