Hoffman constant of the argmin mapping in linear optimization

TL;DR

This paper derives the first exact formula for the Hoffman constant of the argmin mapping in linear optimization, using a novel approach involving well-connected piecewise convex mappings.

Contribution

It introduces a new point-based expression for the Hoffman constant in linear optimization and develops tools like well-connected piecewise convex mappings.

Findings

First exact formula for the Hoffman constant in linear optimization

Establishes a link between the Hoffman constant and calmness moduli

Provides insights into directional stability of solutions

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.