About Error Bounds in Metrizable Topological Vector Spaces

Abstract

This paper aims to present some sufficient criteria under which a given function f : X → Y satisfies the error bound property, where X and Y are either topological vector spaces whose topologies are generated by metrics or metrizable subsets of some topological vector spaces. Then, we discuss the Hoffman estimation and obtain some results for the estimate of the distance to the set of solutions to a system of linear equalities. The advantage of our estimate is that it allows to calculate the coefficient of the error bound. The applications of this presentation are illustrated by some examples.

Primal Characterizations of Stability of Error Bounds for Semi-infinite Convex Constraint Systems in Banach Spaces

“…Research on error bounds can be traced back to Hoffman's pioneering work in [1]. More precisely, given a m × n matrix A and a m-vector b, in order to estimate the Euclidean distance from a point x to its nearest point in the polyhedral set {u : Au ≤ b}, Hoffman [1] showed that this distance is bounded from above by some constant (only dependent on A) times the Euclidean norm of the residual error (Ax − b) + , where (Ax − b) + denotes the positive part of Ax − b. Hoffman's work has been well-recognized and extensively studied and extended by many authors, including Robinson [2], Mangasarian [3], Auslender & Crouzeix [4], Pang [5], Lewis & Pang [6], Klatte & Li [7], Jourani [8] , Abassi & Théra [9,10], Ioffe [11] and many others. The abundant literature on error bounds shows that this theory has important applications in the sensitivity analysis of linear/integer programs (cf.…”