Perturbations and Metric Regularity

Author: Dontchev A.   Lewis A.  

Publisher: Springer Publishing Company

ISSN: 0927-6947

Source: Set-Valued Analysis, Vol.13, Iss.4, 2005-12, pp. : 417-438

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Abstract

A point x is an approximate solution of a generalized equation bF(x) if the distance from the point b to the set F(x) is small. ‘Metric regularity’ of the set-valued mapping F means that, locally, a constant multiple of this distance bounds the distance from x to an exact solution. The smallest such constant is the ‘modulus of regularity’, and is a measure of the sensitivity or conditioning of the generalized equation. We survey recent approaches to a fundamental characterization of the modulus as the reciprocal of the distance from F to the nearest irregular mapping. We furthermore discuss the sensitivity of the regularity modulus itself, and prove a version of the fundamental characterization for mappings on Riemannian manifolds.