New characterizations of Hoffman constants for systems of linear constraints

Abstract

We give a characterization of the Hoffman constant of a system of linear constraints in ${\mathbb{R}}^n$ {\em relative} to a {\em reference polyhedron} $R\subseteq {\mathbb{R}}^n$. The reference polyhedron $R$ represents constraints that are easy to satisfy such as box constraints. In the special case $R={\mathbb{R}}^n$, we obtain a novel characterization of the classical Hoffman constant. More precisely, suppose $R\subseteq {\mathbb{R}}^n$ is a reference polyhedron, $A\in {\mathbb{R}}^{m\times n},$ and $A(R):=\{Ax:x\in R\}$. We characterize the sharpest constant $H(A|R)$ such that for all $b\in A(R)+{\mathbb{R}}_{+}^m$ and $u\in R$ \[ \dist(u, P_{A}(b)\cap R) \le H(A|R) \cdot \|(Au-b)_+\|, \] where $P_A(b)=\{x\in {\mathbb{R}}^n:Ax\le b\}$. Our characterization is stated in terms of the largest of a canonical collection of easily computable Hoffman constants. Our characterization in turn suggests new algorithmic procedures to compute Hoffman constants.

30 pages. To Appear in Mathematical Programming. arXiv admin note: text overlap with arXiv:1804.08418