# Metric subregularity for nonclosed convex multifunctions in normed spaces

ESAIM: Control, Optimisation and Calculus of Variations (2010)

- Volume: 16, Issue: 3, page 601-617
- ISSN: 1292-8119

## Access Full Article

top## Abstract

top## How to cite

topZheng, Xi Yin, and Ng, Kung Fu. "Metric subregularity for nonclosed convex multifunctions in normed spaces." ESAIM: Control, Optimisation and Calculus of Variations 16.3 (2010): 601-617. <http://eudml.org/doc/250749>.

@article{Zheng2010,

abstract = {
In terms of the normal cone and the coderivative,
we provide some necessary and/or sufficient conditions of metric subregularity for
(not necessarily closed) convex multifunctions in normed spaces. As applications, we present some
error bound results for (not necessarily lower semicontinuous) convex functions on normed
spaces. These results improve and extend some existing error bound results.
},

author = {Zheng, Xi Yin, Ng, Kung Fu},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Metric subregularity; multifunction; normal cone; coderivative; metric subregularity; convex multifunction},

language = {eng},

month = {7},

number = {3},

pages = {601-617},

publisher = {EDP Sciences},

title = {Metric subregularity for nonclosed convex multifunctions in normed spaces},

url = {http://eudml.org/doc/250749},

volume = {16},

year = {2010},

}

TY - JOUR

AU - Zheng, Xi Yin

AU - Ng, Kung Fu

TI - Metric subregularity for nonclosed convex multifunctions in normed spaces

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2010/7//

PB - EDP Sciences

VL - 16

IS - 3

SP - 601

EP - 617

AB -
In terms of the normal cone and the coderivative,
we provide some necessary and/or sufficient conditions of metric subregularity for
(not necessarily closed) convex multifunctions in normed spaces. As applications, we present some
error bound results for (not necessarily lower semicontinuous) convex functions on normed
spaces. These results improve and extend some existing error bound results.

LA - eng

KW - Metric subregularity; multifunction; normal cone; coderivative; metric subregularity; convex multifunction

UR - http://eudml.org/doc/250749

ER -

## References

top- J.F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems. Springer, New York (2000).
- J.V. Burke and S. Deng, Weak sharp minima revisited, Part I: Basic theory. Control Cybern.31 (2002) 399–469.
- J.V. Burke and S. Deng, Weak sharp minima revisited, Part III: Error bounds for differentiable convex inclusions. Math. Program.116 (2009) 37–56.
- P.L. Combettes, Strong convergence of block-iterative outer approximation methods for convex optimzation. SIAM J. Control Optim.38 (2000) 538–565.
- A.L. Dontchev and R.T. Rockafellar, Regularity and conditioning of solution mappings in variational analysis. Set-Valued Anal.12 (2004) 79–109.
- A.L. Dontchev, A.S. Lewis and R.T. Rockafellar, The radius of metric regularity. Trans. Amer. Math. Soc.355 (2003) 493–517.
- R. Henrion and A. Jourani, Subdifferential conditions for calmness of convex constraints. SIAM J. Optim.13 (2002) 520–534.
- R. Henrion and J. Outrata, Calmness of constraint systems with applications. Math. Program.104 (2005) 437–464.
- R. Henrion, A. Jourani and J. Outrata, On the calmness of a class of multifunctions. SIAM J. Optim.13 (2002) 603–618.
- H. Hu, Characterizations of the strong basic constraint qualification. Math. Oper. Res.30 (2005) 956–965.
- H. Hu, Characterizations of local and global error bounds for convex inequalities in Banach spaces. SIAM J. Optim.18 (2007) 309–321.
- A.D. Ioffe, Metric regularity and subdifferential calculus. Russian Math. Surveys55 (2000) 501–558.
- D. Klatte and B. Kummer, Nonsmooth Equations in Optimization, Regularity, Calculus, Methods and Applications; Nonconvex Optimization and its Application60. Kluwer Academic Publishers, Dordrecht (2002).
- A. Lewis and J.S. Pang, Error bounds for convex inequality systems, in Generalized Convexity, Generalized Monotonicity: Recent Results, Proceedings of the Fifth Symposium on Generalized Convexity, Luminy, June 1996, J.-P. Crouzeix, J.-E. Martinez-Legaz and M. Volle Eds., Kluwer Academic Publishers, Dordrecht (1997) 75–100.
- W. Li, Abadie's constraint qualification, metric regularity, and error bounds for differentiable convex inequalities. SIAM J. Optim.7 (1997) 966–978.
- W. Li and I. Singer, Global error bounds for convex multifunctions and applications. Math. Oper. Res.23 (1998) 443–462.
- B.S. Mordukhovich, Complete characterization of openness, metric regularity, and Lipschitzian properties of multifunctions. Trans. Amer. Math. Soc.340 (1993) 1–35.
- K.F. Ng and X.Y. Zheng, Error bound for lower semicontinuous functions in normed spaces. SIAM J. Optim.12 (2001) 1–17.
- S.M. Robinson, Regularity and stability for convex multivalued fucntions. Math. Oper. Res.1 (1976) 130–143.
- C. Zalinescu, Weak sharp minima, well-behaving functions and global error bounds for convex inequalities in Banach spaces, in Proc. 12th Baical Internat. Conf. on Optimization Methods and their applications, Irkutsk, Russia (2001) 272–284.
- C. Zalinescu,Convex Analysis in General Vector Spaces. World Scientific, Singapore (2002).
- C. Zalinescu, A nonlinear extension of Hoffman's error bounds for linear inequalities. Math. Oper. Res.28 (2003) 524–532.
- X.Y. Zheng and K.F. Ng, Metric regularity and constraint qualifications for convex inequalities on Banach spaces. SIAM J. Optim.14 (2003) 757–772.
- X.Y. Zheng and K.F. Ng, Metric subregularity and constraint qualifications for convex generalized equations in Banach spaces. SIAM. J. Optim.18 (2007) 437–460.
- X.Y. Zheng and K.F. Ng, Linear regularity for a collection of subsmooth sets in Banach spaces. SIAM J. Optim.19 (2008) 62–76.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.