A Unified Analysis of Hoffman's Bound via Fenchel Duality

In 1952, Hoffman showed that the distance from any point to the solution set of a linear system is bounded above by a constant times the norm of the residual. Subsequently, this bound has been studied extensively and has found many practical uses. In this paper, we consider an extension of Hoffman’s bound to a partially infinite-dimensional setting in which the norm defining the distance is replaced by a positively homogeneous convex function and the linear system is replaced by a convex inclusion of the form $Ax - a \in K$, where A is a continuous linear operator from some normed linear space into $\mathbf{R}^m ,a$ is an element of the range space of A, and K is a nonempty closed convex cone in $\mathbf{R}^m $. When specialized to the finite-dimensional case, we unify and extend many existing results on Hoffman’s bound. Our analysis is based on the use of Fenchel duality to express the distance as the supremum of a certain concave function over a bounded subset of the polar of K. Much of our analysis als...