A stroll in the jungle of error bounds

Abstract

The aim of this paper is to give a short overview on error bounds and to provide the first bricks of a unified theory. Inspired by the works of [8,15,13,16, 10], we show indeed the centrality of the Lojasiewicz gradient inequality. For this, we review some necessary and sufficient conditions for global/local error bounds, both in the convex and nonconvex case. We also recall some results on quantitative error bounds which play a major role in convergence rate analysis and complexity theory of many optimization methods.

New characterizations of Hoffman constants for systems of linear constraints

“…The bound (1) is a type of error bound for the system of inequalities Ax ≤ b, that is, an inequality bounding the distance from a point u ∈ R n to a nonempty solution set in terms of a measure of the error or residual of the point u. The Hoffman bound (1) and more general error bounds play a fundamental role in mathematical programming [30,31,46]. In particular, Hoffman bounds as well as other related error bounds are instrumental in establishing convergence properties of a variety of algorithms [4,13,17,21,22,26,28,35,43].…”

“…Observe that A J (R n ) + R J + = R J if and only if A J x < 0 is feasible. Hoffman bounds of the classical form (1), the relative form (2), and more general error bounds play a fundamental role in mathematical programming [29,30,51]. In particular, these kinds of Hoffman bounds as well as other related error bounds play a central role in establishing convergence properties of a variety of modern convex optimization algorithms [4,11,14,19,20,26,28,33,47].…”