New characterizations of Hoffman constants for systems of linear constraints

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