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Perturbations and metric regularity

A.L. Dontchev (ald***at***ams.org)
A.S. Lewis (aslewis***at***orie.cornell.edu)

Abstract: A point x is an approximate solution of a generalized equation [b lies in F(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.

Keywords: metric regularity, set-valued mapping, condition number, Eckart-Young theorem, distance to ill-posedness, Riemannian manifold, regularity modulus.

Category 1: Convex and Nonsmooth Optimization (Nonsmooth Optimization )

Citation: Technical Report, ORIE, Cornell University

Download: [PDF]

Entry Submitted: 12/15/2004
Entry Accepted: 12/15/2004
Entry Last Modified: 12/15/2004

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