An easily computable upper bound on the Hoffman constant for homogeneous inequality systems

TL;DR

This paper introduces a simple, computationally feasible method to estimate an upper bound on the Hoffman constant for homogeneous systems, which is typically difficult to compute.

Contribution

It provides a tractable and easily implementable procedure to compute an upper bound on the Hoffman constant for homogeneous inequalities.

Findings

The procedure is fully tractable and implementable.

It offers a practical way to estimate the Hoffman constant.

The method contrasts with the intractability of computing general Hoffman constants.

Abstract

Let $A\in \mathbb{R}^{m\times n}\setminus \{0\}$ and $P:=\{x:Ax\le 0\}$. This paper provides a procedure to compute an upper bound on the following homogeneous Hoffman constant: \[ H_0(A) := \sup_{u\in \mathbb{R}^n \setminus P} \frac{\text{dist}(u,P)}{\text{dist}(Au, \mathbb{R}^m_-)}. \] In sharp contrast to the intractability of computing more general Hoffman constants, the procedure described in this paper is entirely tractable and easily implementable.