We give a characterization of the Hoffman constant of a system of linear constraints in $\R^n$ relative to a 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, given a reference polyhedron $R\subseteq \R^n$ and $A\in \R^{m\times n}$, we characterize the sharpest constant $H(A\vert R)$ such that for all $b \in A(R) + \R^m_+$ and $u\in R$ \[ \dist(u, P_{A}(b)\cap R) \le H(A\vert R) \cdot \|(Au-b)_+\|, \] 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.
Citation
Working Paper, Tepper School of Business, Carnegie Mellon University, May 2019