From Calmness to Hoffman Constants for Linear Semi-infinite Inequality Systems

Abstract

In this paper we focus on different---global, semilocal, 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 sides. The Hoffman modulus is shown to coincide with the Lipschitz upper semicontinuity modulus and the supremum of calmness moduli when confined to multifunctions with a convex graph and closed images in a reflexive Banach space, which is the case for our feasible set mapping. Moreover, for this particular multifunction a formula---involving only 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 (semilocal) 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.