Perturbations and Metric Regularity

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.

[1]  R. Tyrrell Rockafellar,et al.  Variational Analysis , 1998, Grundlehren der mathematischen Wissenschaften.

[2]  A. S. Lewis,et al.  Ill-Conditioned Convex Processes and Conic Linear Systems , 1999, Math. Oper. Res..

[3]  Felipe Cucker,et al.  A note on level-2 condition numbers , 2005, J. Complex..

[4]  S. M. Robinson.  Normed convex processes , 1972 .

[5]  W. Boothby.  An introduction to differentiable manifolds and Riemannian geometry , 1975 .

[6]  Andrei Dmitruk,et al.  LYUSTERNIK'S THEOREM AND THE THEORY OF EXTREMA , 1980 .

[7]  Yu. S. Ledyaev,et al.  Nonsmooth analysis on smooth manifolds , 2007 .

[8]  Boris S. Mordukhovich.  Coderivative Analysis of Variational Systems , 2004, J. Glob. Optim..

[9]  T. Zolezzi.  On the Distance Theorem in Quadratic Optimization , 2002 .

[10]  J. Demmel.  On condition numbers and the distance to the nearest ill-posed problem , 2015 .

[11]  J. Nash.  The imbedding problem for Riemannian manifolds , 1956 .

[12]  Michael E. Taylor,et al.  Differential Geometry I , 1994 .

[14]  Alexander D. Ioffe.  On Perturbation Stability of Metric Regularity , 2001 .

[15]  A. D. Ioffe.  On stability estimates for the regularity property of maps , 2003 .

[16]  Adrian S. Lewis,et al.  Ill-Conditioned Convex Processes and Linear Inequalities , 1998 .

[17]  Asen L. Dontchev,et al.  Regularity and Conditioning of Solution Mappings in Variational Analysis , 2004 .

[18]  Lawrence M. Graves,et al.  Some mapping theorems , 1950 .

[19]  F. Clarke.  Optimization And Nonsmooth Analysis , 1983 .

[20]  Александр Давидович Иоффе,et al.  Метрическая регулярность и субдифференциальное исчисление@@@Metric regularity and subdifferential calculus , 2000 .

[21]  A. Aleksandrov,et al.  Intrinsic Geometry of Surfaces , 1967 .

[22]  A. Ioffe.  Metric regularity and subdifferential calculus , 2000 .

[23]  R. Rockafellar,et al.  The radius of metric regularity , 2002 .

[24]  Marco A. López,et al.  Metric regularity of semi-infinite constraint systems , 2005, Math. Program..

[25]  S. Helgason.  Differential Geometry, Lie Groups, and Symmetric Spaces , 1978 .

[26]  Alexander D. Ioffe.  On robustness of the regularity property of maps , 2003 .