Error bounds for solutions of linear equations and inequalities

Abstract

Given a system of linear equations and inequalities in n variables, a famous result due to A. J. Hoffman (1952) says that the distance of any point in R" to the solution set of this system is bounded above by the product of a positive constant and the absolute residual. We shall discuss explicit representations of this constant in dependence upon the pair of norms used for the estimation. A method for computing a special form of Hoffman constants is proposed. Finally, we use these results in the analysis of Lipsehitz continuity for solutions of parametric quadratic programs.

The positiveness of lower limits of the Hoffman constant in parametric polyhedral programs

“…then the assumptions of the above proposition are satisfied, where f is defined in (14) and cl * stands for the closure with respect to the weak * topology. Indeed, because of the equi-continuity of (a * i (·)), there exists a neighborhood U (t 0 ) of t 0 such that…”

“…Many authors have presented and studied explicit representations of Hoffman constants, we refer to [3,4,6,7,10,[14][15][16][17]23,28,29] and references therein, see also [19]. In [7,Theorem 5.1] (under the assumption K (t) = ∅) it is shown that…”

“…Namely, in what we do herein is employment of an approximate technique. We approximate subgradients of the convex function defined in (14) by subgradients of functions…”

Comments on: Stability in linear optimization and related topics. A personal tour

“…Numerical experiments for computing sharp Lipschitz constants are not much available. In our context, Klatte and Thiere (1995) discuss several concrete representations of ν α2 (A, C), i.e., for · β = · 2 . In particular, one derives for · α = · ∞ (maximum norm) and · β = · 2 that after some transformations the computation of ν ∞2 consists in solving a finite number of special quadratic programs of the form min{…”

“…where A i and C j are the rows of A and C. Thiere's tests done in the late 1980ies, see Klatte and Thiere (1995), were never repeated by using more recent generations of solvers. Since the computable formulae for exact Lipschitz bounds under metric regularity (see the author's Section 2.3 and e.g.…”