Perturbations and Metric Regularity |
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| Authors: | A L Dontchev and A S Lewis |
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| Institution: | (1) Mathematical Reviews, American Mathematical Society, Ann Arbor, MI, 48107-8604, U.S.A;(2) School of ORIE, Cornell University, Ithaca, NY, 14853, U.S.A |
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| Abstract: | A point x is an approximate solution of a generalized equation b∈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.
Mathematics Subject Classifications 2000 Primary: 49J53; secondary: 90C31. |
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| Keywords: | metric regularity set-valued mapping condition number Eckart– Young theorem distance to ill-posedness Riemannian manifold regularity modulus |
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