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

Let $$A\in {\mathbb R}^{m\times n}\setminus \{0\}$$ A ∈ R m × n \ { 0 } and $$P:=\{x:Ax\le 0\}$$ P : = { x : A x ≤ 0 } . This paper provides a procedure to compute an upper bound on the following homogeneous Hoffman constant $$\begin{aligned} H_0(A):= \sup _{u\in {\mathbb R}^n \setminus P} \frac{{{\,\textrm{dist}\,}}(u,P)}{{{\,\textrm{dist}\,}}(Au, {\mathbb R}^m_-)}. \end{aligned}$$ H 0 ( A ) : = sup u ∈ R n \ P dist ( u , P ) dist ( A u , 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.