  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.
