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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A simple closure condition for the normal cone intersection formula


Authors: Regina Sandra Burachik and Vaithilingam Jeyakumar
Journal: Proc. Amer. Math. Soc. 133 (2005), 1741-1748
MSC (2000): Primary 46N10, 90C25
Published electronically: December 21, 2004
MathSciNet review: 2120273
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Abstract: In this paper it is shown that if $C$ and $D$ are two closed convex subsets of a Banach space $X$ and $x\in C\cap D$, then $N_{C\cap D}(x)=N_{C}(x)+N_{D}(x)$ whenever the convex cone, $\left(\mathrm{Epi} \,\sigma _{C}+\mathrm{Epi}\,\sigma _{D}\right)$, is weak* closed, where $ \sigma _{C}$ and $N_{C}$ are the support function and the normal cone of the set $C$ respectively. This closure condition is shown to be weaker than the standard interior-point-like conditions and the bounded linear regularity condition.


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Additional Information

Regina Sandra Burachik
Affiliation: Engenharia de Sistemas e Computacao, COPPE-UFRJ CP 68511, Rio de Janeiro-RJ, CEP 21945-970, Brazil
Email: regi@cos.ufrj.br

Vaithilingam Jeyakumar
Affiliation: Department of Applied Mathematics, University of New South Wales, Sydney 2052, Australia
Email: jeya@maths.unsw.edu.au

DOI: http://dx.doi.org/10.1090/S0002-9939-04-07844-X
PII: S 0002-9939(04)07844-X
Keywords: Normal cone, closure condition, bounded linear regularity, convex optimization, strong conical hull intersection property
Received by editor(s): February 16, 2004
Published electronically: December 21, 2004
Additional Notes: The first author’s research was supported by CAPES (Grant BEX 0664-02/2), and was partially completed while the author was a visitor at the School of Mathematics, University of New South Wales, Sydney, and the School of Mathematics and Statistics, University of South Australia. This author wishes to thank the School of Mathematics at the University of South Australia for providing a stimulating environment and good infrastructure during her longer stay there, and also to the School of Mathematics at the University of New South Wales for their support. The authors are thankful to Jonathan Borwein for his helpful suggestions and for referring us to paper [4] and to the referee for the careful reading of the manuscript.
Communicated by: Jonathan M. Borwein
Article copyright: © Copyright 2004 American Mathematical Society