Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-04-07844-X
Published electronically: December 21, 2004
MathSciNet review: 2120273
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • 1. S. Adly, E. Emil and M. Thera, On the closedness of the algebraic difference of closed convex sets, J. Math. Pures Appl., 82(9) (2003), 1219-1249. MR 2012809 (2004g:49028)
  • 2. H. Attouch and H. Brezis, Duality for the sum of convex functions in general Banach spaces, in Aspects of Mathematics and Its Applications, J. A. Barroso, ed., Elsevier, Amsterdam, 1986, 125-133. MR 0849549 (87m:90095)
  • 3. H. H. Bauschke and J. M. Borwein, On projection algorithms for solving convex feasibility problems, SIAM. Rev., 38 (1993), 367-426. MR 1409591 (98f:90045)
  • 4. H. H. Bauschke, J. M. Borwein and W. Li, Strong conical hull intersection property, bounded linear regularity, Jameson's property (G), and error bounds in convex optimization, Math. Progr., 86 (1999), 135-160. MR 1712477 (2000f:90095)
  • 5. H. H. Bauschke, J. M. Borwein and P. Tseng, Bounded linear regularity, strong CHIP, and CHIP are distinct properties, J. Convex Analysis, 7(2) (2000), 395-412. MR 1811687 (2002d:90113)
  • 6. J. M. Borwein and H. Wolkowicz, A simple constraint qualification in infinite dimensional programming, Math. Progr., 35 (1986), 83 - 96. MR 0842636 (87i:90321)
  • 7. M. Ciligot-Travain, An intersection formula for the normal cone associated with the hypertangent cone, Appl. Anal., 5 (1999), 239-247. MR 1722221 (2000h:49025)
  • 8. M. Cotlar and R. Cignoli, An Introduction to Functional Analysis, North-Holland Publishing Company, The Netherlands, 1974. MR 0405049 (53:8845)
  • 9. F. Deutsch, W. Li and J. D. Ward, Best approximation from the intersection of a closed convex set and a polyhedron in Hilbert space, weak Slater conditions, and the strong conical hull intersection property, SIAM J. Optim., 10 (1999), 252-268. MR 1742319 (2001a:41047)
  • 10. M.S. Gowda and M. Teboulle, A comparison of constraint qualifications in infinite-dimensional convex programming, SIAM J. Control and Optim., 28 (4), 1990, 925 - 935. MR 1051630 (91f:90100)
  • 11. J. B. Hiriart-Urruty and C. Lemarechal, Convex Analysis and Minimization Algorithms, Volumes I and II, Springer-Verlag, Berlin Heidleberg, 1993. MR 1261420 (95m:90001) MR 1295240 (95m:90002)
  • 12. R. B. Holmes, Geometric Functional Analysis and its Applications, Springer-Verlag, Berlin, 1975. MR 0410335 (53:14085)
  • 13. V. Jeyakumar, Duality in infinite dimensional optimization, Nonlinear Anal. T,M& A, 15 (1990), 1111-1122. MR 1082286 (91k:49053)
  • 14. V. Jeyakumar, G. M. Lee and N. Dinh, New sequential Lagrange multiplier conditions characterizing optimality without constraint qualification for convex programs, SIAM J. Optim., 14(2) (2003), 534-547. MR 2048156
  • 15. V. Jeyakumar and H. Wolkowicz, Generalizations of Slater's constraint qualification for infinite convex programs, Math. Progr., 57(1) (1992), 85-102. MR 1167408 (93e:90070)
  • 16. F. Jules and M. Lassonde, Formulas for subdifferentials of sums of convex functions, J. Convex Anal., 9(2), (2002), 519-533. MR 1970570 (2004b:49029)
  • 17. C. Li and X. Jin, Nonlinearly constrained best approximation in Hilbert spaces: the strong CHIP, and the basic constraint qualification, SIAM J. Optim., 13(1) (2002), 228-239. MR 1922763 (2003k:41048)
  • 18. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, (1970), NJ. MR 0274683 (43:445)
  • 19. T. Strömberg, The operation of infimal convolution, Diss. Math., 352, (1996), 1-61. MR 1387951 (97c:49018)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 46N10, 90C25

Retrieve articles in all journals with MSC (2000): 46N10, 90C25


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: https://doi.org/10.1090/S0002-9939-04-07844-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

American Mathematical Society