Advertisement

Journal of Global Optimization

, Volume 57, Issue 3, pp 663–676 | Cite as

Projections onto convex sets on the sphere

Article

Abstract

In this paper some concepts of convex analysis are extended in an intrinsic way from the Euclidean space to the sphere. In particular, relations between convex sets in the sphere and pointed convex cones are presented. Several characterizations of the usual projection onto a Euclidean convex set are extended to the sphere and an extension of Moreau’s theorem for projection onto a pointed convex cone is exhibited.

Keywords

Sphere Pointed convex cone Convex set in the sphere Projection onto a pointed convex cone 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Dahl G., Leinaas J.M., Myrheim J., Ovrum E.: A tensor product matrix approximation problem in quantum physics. Linear Algebra Appl. 420(2–3), 711–725 (2007)CrossRefGoogle Scholar
  2. 2.
    Das P., Chakraborti N.R., Chaudhuri P.K.: Spherical minimax location problem. Comput. Optim. Appl. 18(3), 311–326 (2001)CrossRefGoogle Scholar
  3. 3.
    do Carmo, M.P.: Differential Geometry of Curves and Surfaces. Prentice-Hall Inc., Englewood Cliffs, N.J. (Translated from the Portuguese) (1976)Google Scholar
  4. 4.
    do Carmo, M.P.: Riemannian Geometry. Mathematics: Theory and Applications. Birkhäuser Boston Inc., Boston, MA (Translated from the second Portuguese edition by Francis Flaherty) (1992)Google Scholar
  5. 5.
    Drezner Z., Wesolowsky G.O.: Minimax and maximin facility location problems on a sphere. Naval Res. Logist. Q. 30(2), 305–312 (1983)CrossRefGoogle Scholar
  6. 6.
    Han D., Dai H.H., Qi L.: Conditions for strong ellipticity of anisotropic elastic materials. J. Elast. 97(1), 1–13 (2009)CrossRefGoogle Scholar
  7. 7.
    Iusem A., Seeger A.: On pairs of vectors achieving the maximal angle of a convex cone. Math. Program. Ser. B 104(2–3), 501–523 (2005)CrossRefGoogle Scholar
  8. 8.
    Iusem A., Seeger A.: Searching for critical angles in a convex cone. Math. Program. Ser. B 120(1), 3–25 (2009)CrossRefGoogle Scholar
  9. 9.
    Moreau J.-J.: Décomposition orthogonale d’un espace hilbertien selon deux cônes mutuellement polaires. C. R. Acad. Sci. Paris 255, 238–240 (1962)Google Scholar
  10. 10.
    Qi L.: Eigenvalues of a real supersymmetric tensor. J. Symb. Comput. 40(6), 1302–1324 (2005)CrossRefGoogle Scholar
  11. 11.
    Qi L., Teo K.L.: Multivariate polynomial minimization and its application in signal processing. J. Glob. Optim. 26(4), 419–433 (2003)CrossRefGoogle Scholar
  12. 12.
    Sakai, T.: Riemannian Geometry, Vol. 149 of Translations of Mathematical Monographs. American Mathematical Society, Providence, RI (Translated from the 1992 Japanese original by the author) (1996)Google Scholar
  13. 13.
    Smith, S.T.: Optimization techniques on Riemannian manifolds. In: Bloch, A. (ed.) Hamiltonian and Gradient Flows, Algorithms and Control, vol. 3 of Fields Inst. Commun., pp. 113–136. Am. Math. Soc., Providence, RI (1994)Google Scholar
  14. 14.
    Walter R.: On the metric projection onto convex sets in riemannian spaces. Arch. Math. (Basel) 25, 91–98 (1974)CrossRefGoogle Scholar
  15. 15.
    Weiland S., van Belzen F.: Singular value decompositions and low rank approximations of tensors. IEEE Trans. Signal Process. 58(3, part 1), 1171–1182 (2010)CrossRefGoogle Scholar
  16. 16.
    Xue G.-L.: A globally convergent algorithm for facility location on a sphere. Comput. Math. Appl. 27(6), 37–50 (1994)CrossRefGoogle Scholar
  17. 17.
    Xue G.L.: On an open problem in spherical facility location. Numer. Algorithms 9(1–2), 1–12 (1995)CrossRefGoogle Scholar
  18. 18.
    Xue G.L.: Algorithms for Constrained Approximation and Optimization. Stowe, VT (1993)Google Scholar
  19. 19.
    Zhang L.: On the convergence of a modified algorithm for the spherical facility location problem. Oper. Res. Lett. 31(2), 161–166 (2003)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC. 2012

Authors and Affiliations

  1. 1.IME/UFGGoiâniaBrazil
  2. 2.Instituto de Matemática Pura e AplicadaRio de JaneiroBrazil
  3. 3.School of MathematicsThe University of BirminghamBirminghamUK

Personalised recommendations