Abstract

The existence of continuous isotone retractions onto pointed closed convex cones in Hilbert spaces is studied. The cones admitting such mappings are called isotone retraction cones. In finite dimension, generating, isotone retraction cones are polyhedral. For a closed, pointed, generating cone in a Hilbert space the isotonicity of a retraction and its complement implies that the cone is latticial and the retraction is well defined by the latticial structure. The notion of sharp mapping is introduced. If the cone is generating and normal, it is proved that its latticiality is equivalent to the existence of an isotone retraction onto it, whose complement is sharp. The subdual and autodual latticial cones are also characterized by isotonicity. This is done by attempting to extend Moreau's theorem to retractions

Author information

Contact details are reproduced from the original publication and may be historical.

Sandor Z. Németh

School of Mathematics, University of Birmingham, The Watson Building, Edgbaston - Birmingham B15 2TT, England

nemeths@for.mat.bham.ac.uk

Alexandru B. Németh

Faculty of Mathematics and Computer Science, Babeş-Bolyai University, Mihail Kogălniceanu Nr. 1, 400084 Cluj-Napoca, Romania

nemab@math.ubbcluj.ro

References

  1. G. Isac, A. B. Németh: Monotonicity of metric projections onto positive cones of ordered Euclidean spaces, Arch. Math. 46(6) (1986) 568–576.https://doi.org/10.1007/BF01195027Google Scholar
  2. G. Isac, A. B. Németh: Every generating isotone projection cone is latticial and correct, J. Math. Anal. Appl. 147(1) (1990) 53–62.https://doi.org/10.1016/0022-247X(90)90383-QGoogle Scholar
  3. S. J. Bernau: Isotone projection cones, in: Ordered Algebraic Structures (Gainesville, 1991), J. Martinez et al. (ed.), Kluwer, Dordrecht (1993) 3–10.https://doi.org/10.1007/978-94-011-1723-4_1Google Scholar
  4. G. Isac, A. B. Németh: Isotone projection cones in Hilbert spaces and the complementarity problem, Boll. Unione Mat. Ital., VII. Ser., B 4(4) (1990) 773–802.Google Scholar
  5. G. Isac, A. B. Németh: Projection methods, isotone projection cones, and the complementarity problem, J. Math. Anal. Appl. 153(1) (1990) 258–275.https://doi.org/10.1016/0022-247X(90)90277-MGoogle Scholar
  6. G. Isac, S. Z. Németh: Regular exceptional family of elements with respect to isotone projection cones in Hilbert spaces and complementarity problems, Optim. Lett. 2(4) (2008) 567–576.https://doi.org/10.1007/s11590-008-0083-8Google Scholar
  7. S. Z. Németh: Iterative methods for nonlinear complementarity problems on isotone projection cones, J. Math. Anal. Appl. 350(1) (2009) 340–347.https://doi.org/10.1016/j.jmaa.2008.09.066Google Scholar
  8. S. Z. Németh: Isotone retraction cones in Hilbert spaces, Nonlinear Anal. 73(2) (2010) 495–499.https://doi.org/10.1016/j.na.2010.03.044Google Scholar
  9. S. Z. Németh: Characterization of latticial cones in Hilbert spaces by isotonicity and generalized infimum, Acta Math. Hung. 127(4) (2010) 376–390.https://doi.org/10.1007/s10474-009-9145-3Google Scholar
  10. J.-J. Moreau: Décomposition orthogonale d’un espace hilbertien selon deux cônes mutuellement polaires, C. R. Acad. Sci., Paris 255 (1962) 238–240.Google Scholar
  11. C. W. McArthur: In what spaces is every closed normal cone regular?, Proc. Edinb. Math. Soc., II. Ser. 17 (1970) 121–125.https://doi.org/10.1017/S001309150000938XGoogle Scholar
  12. S. Straszewitz: Über exponierte Punkte abgeschlossener Punktmengen, Fundam. Math. 24 (1935) 139–143.https://doi.org/10.4064/fm-24-1-139-143Google Scholar

Suggested citation

S. Z. Németh, A. B. Németh. “About the Existence of an Isotone Retraction onto a Convex Cone.” Journal of Convex Analysis 18 (2011), No. 3, 707–720. https://doi.org/10.68381/jca18045

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2011. Rights now held by Banach Press.