Hoffman constant of the argmin mapping in linear optimization

Abstract

The main goal of this paper is to provide a point-based expression for the Hoffman constant of the argmin mapping in linear optimization, understood as the sharp Lipschitz constant restricted to its domain. The work is mainly developed in the parametric context of right-hand side perturbations of the constraint system. To the authors' knowledge, this is the first exact formula for this constant, although we can find in the literature different upper estimates. The paper tackles this objective from a broader perspective, which introduces new tools of their own interest, such as the concept of well-connected piecewise convex mapping. We isolate the nice behavior of such mappings to derive a crucial equality between the Hoffman constant (which is a global stability measure) and the supremum of calmness moduli (of local nature). The paper also includes some specifics about directional stability of optimal solutions and finishes with some conclusions and notes about further research.