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Metric subregularity for nonclosed convex multifunctions innormed spaces

Published online by Cambridge University Press:  18 June 2009

Xi Yin Zheng  and
Kung Fu Ng
Xi Yin Zheng
Affiliation:
Department of Mathematics, Yunnan University, Kunming 650091, P. R. China. xyzheng@ynu.edu.cn
Kung Fu Ng
Affiliation:
Department of Mathematics, The Chinese University of Hong Kong, Shatin, New Territory, Hong Kong. kfng@math.cuhk.edu.hk
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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 someerror bound results for (not necessarily lower semicontinuous) convex functions on normedspaces. These results improve and extend some existing error bound results.

Keywords

Metric subregularitymultifunctionnormal conecoderivative
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 16 , Issue 3 , July 2010 , pp. 601 - 617
Copyright
© EDP Sciences, SMAI, 2009

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References

J.F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems. Springer, New York (2000).
Burke, J.V. and Deng, S., Weak sharp minima revisited, Part I: Basic theory. Control Cybern. 31 (2002) 399–469.
Burke, J.V. and Deng, S., Weak sharp minima revisited, Part III: Error bounds for differentiable convex inclusions. Math. Program. 116 (2009) 37–56. CrossRef
Combettes, P.L., Strong convergence of block-iterative outer approximation methods for convex optimzation. SIAM J. Control Optim. 38 (2000) 538–565. CrossRef
Dontchev, A.L. and Rockafellar, R.T., Regularity and conditioning of solution mappings in variational analysis. Set-Valued Anal. 12 (2004) 79–109. CrossRef
Dontchev, A.L., Lewis, A.S. and Rockafellar, R.T., The radius of metric regularity. Trans. Amer. Math. Soc. 355 (2003) 493–517. CrossRef
Henrion, R. and Jourani, A., Subdifferential conditions for calmness of convex constraints. SIAM J. Optim. 13 (2002) 520–534. CrossRef
Henrion, R. and Outrata, J., Calmness of constraint systems with applications. Math. Program. 104 (2005) 437–464. CrossRef
Henrion, R., Jourani, A. and Outrata, J., On the calmness of a class of multifunctions. SIAM J. Optim. 13 (2002) 603–618.
Characterizations, H. Hu of the strong basic constraint qualification. Math. Oper. Res. 30 (2005) 956–965.
Characterizations, H. Hu of local and global error bounds for convex inequalities in Banach spaces. SIAM J. Optim. 18 (2007) 309–321.
Ioffe, A.D., Metric regularity and subdifferential calculus. Russian Math. Surveys 55 (2000) 501–558. CrossRef
D. Klatte and B. Kummer, Nonsmooth Equations in Optimization, Regularity, Calculus, Methods and Applications; Nonconvex Optimization and its Application 60. 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.
Abadie's, W. Li constraint qualification, metric regularity, and error bounds for differentiable convex inequalities. SIAM J. Optim. 7 (1997) 966–978.
Li, W. and Singer, I., Global error bounds for convex multifunctions and applications. Math. Oper. Res. 23 (1998) 443–462. CrossRef
Mordukhovich, B.S., Complete characterization of openness, metric regularity, and Lipschitzian properties of multifunctions. Trans. Amer. Math. Soc. 340 (1993) 1–35. CrossRef
Ng, K.F. and Zheng, X.Y., Error bound for lower semicontinuous functions in normed spaces. SIAM J. Optim. 12 (2001) 1–17. CrossRef
Robinson, S.M., Regularity and stability for convex multivalued fucntions. Math. Oper. Res. 1 (1976) 130–143. CrossRef
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).
Zalinescu, C., A nonlinear extension of Hoffman's error bounds for linear inequalities. Math. Oper. Res. 28 (2003) 524–532. CrossRef
Zheng, X.Y. and Metric, K.F. Ng regularity and constraint qualifications for convex inequalities on Banach spaces. SIAM J. Optim. 14 (2003) 757–772. CrossRef
Zheng, X.Y. and Metric, K.F. Ng subregularity and constraint qualifications for convex generalized equations in Banach spaces. SIAM. J. Optim. 18 (2007) 437–460. CrossRef
Zheng, X.Y. and Linear, K.F. Ng regularity for a collection of subsmooth sets in Banach spaces. SIAM J. Optim. 19 (2008) 62–76. CrossRef