From calmness to Hoffman constants for linear semi-infinite inequality systems

TL;DR

This paper investigates Hoffman inequalities for linear semi-infinite systems, establishing relationships between various moduli and deriving formulas for global and local Hoffman constants, with implications for stability analysis.

Contribution

It introduces a unified analysis of Hoffman inequalities in semi-infinite systems, deriving explicit formulas for Hoffman constants based on system properties.

Findings

Hoffman modulus equals the supremum of Lipschitz upper semicontinuity and calmness moduli for convex graph multifunctions.

A formula for the global Hoffman constant involving only the system's left-hand side is derived.

For locally polyhedral systems, a point-based formula for the Hoffman modulus is provided.

Abstract

In this paper we focus on different -- global, semi-local and local -- versions of Hoffman type inequalities expressed in a variational form. In a first stage our analysis is developed for generic multifunctions between metric spaces and we finally deal with the feasible set mapping associated with linear semi-infinite inequality systems (finitely many variables and possibly infinitely many constraints) parameterized by their right-hand side. The Hoffman modulus is shown to coincide with the supremum of Lipschitz upper semicontinuity and calmness moduli when confined to multifunctions with a convex graph and closed images in a reflexive Banach space, which is the case of our feasible set mapping. Moreover, for this particular multifunction a formula -- only involving the system's left-hand side -- of the global Hoffman constant is derived, providing a generalization to our semi-infinite context of finite counterparts developed in the literature. In the particular case of locally polyhedral systems, the paper also provides a point-based formula for the (semi-local) Hoffman modulus in terms of the calmness moduli at certain feasible points (extreme points when the nominal feasible set contains no lines), yielding a practically tractable expression for finite systems.