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In this paper we focus on different -- global, semi-local and local --versions of Hoffman type inequalities expressed in a variational form. In afirst stage our analysis is developed for generic multifunctions between metricspaces and we finally deal with the feasible set mapping associated with linearsemi-infinite inequality systems (finitely many variables and possiblyinfinitely many constraints) parameterized by their right-hand side. TheHoffman modulus is shown to coincide with the supremum of Lipschitz uppersemicontinuity and calmness moduli when confined to multifunctions with aconvex graph and closed images in a reflexive Banach space, which is the caseof our feasible set mapping. Moreover, for this particular multifunction aformula -- only involving the system's left-hand side -- of the global Hoffmanconstant is derived, providing a generalization to our semi-infinite context offinite counterparts developed in the literature. In the particular case oflocally polyhedral systems, the paper also provides a point-based formula forthe (semi-local) Hoffman modulus in terms of the calmness moduli at certainfeasible points (extreme points when the nominal feasible set contains nolines), yielding a practically tractable expression for finite systems.

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