Advertisement

Optimization Letters

, Volume 9, Issue 4, pp 731–741 | Cite as

Projection onto simplicial cones by a semi-smooth Newton method

  • O. P. Ferreira
  • S. Z. Németh
Original Paper

Abstract

In this paper the problem of projection onto a simplicial cone is studied. By using Moreau’s decomposition theorem for projecting onto closed convex cones, the problem of projecting onto a simplicial cone is reduced to finding the unique solution of a nonsmooth system of equations. It is shown that a semi-smooth Newton method applied to the system of equations associated to the problem of projecting onto a simplicial cone is always well defined, and the generated sequence is bounded for any starting point and a formula for any accumulation point of this sequence is presented. It is also shown that under a somewhat restrictive assumption, the semi-smooth Newton method applied to the system of equations associated to the problem of projecting onto a simplicial cone has finite convergence. Besides, under a mild assumption on the simplicial cone, the generated sequence converges linearly to the solution of the associated system of equations.

Keywords

Metric projection Simplicial cones Moreau’s decomposition theorem Semi-smooth Newton method 

Notes

Acknowledgments

The first author was supported in part by FAPEG, CNPq Grants 471815/2012-8, 303732/2011-3 and PRONEX–Optimization(FAPERJ/CNPq). The authors wish to express their gratitude to the reviewers for their helpful comments and for a shorter, more elegant proof of Lemma 5.

References

  1. 1.
    Abbas, M., Németh, S.Z.: Solving nonlinear complementarity problems by isotonicity of the metric projection. J. Math. Anal. Appl. 386(2), 882–893 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Al-Sultan, K.S., Murty, K.G.: Exterior point algorithms for nearest points and convex quadratic programs. Math. Program. 57(2, Ser. B), 145–161 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Bauschke, H.H., Borwein, J.M.: On projection algorithms for solving convex feasibility problems. SIAM Rev. 38(3), 367–426 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Bazaraa, M.S., Sherali, H.D., Shetty, C.M.: Nonlinear Programming. Theory and algorithms, 3rd edn. Wiley, Hoboken (2006)CrossRefzbMATHGoogle Scholar
  5. 5.
    Berk, R., Marcus, R.: Dual cones, dual norms, and simultaneous inference for partially ordered means. J. Am. Statist. Assoc. 91(433), 318–328 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Censor, Y., Elfving, T., Herman, G.T., Nikazad, T.: On diagonally relaxed orthogonal projection methods. SIAM J. Sci. Comput. 30(1), 473–504 (2007/08)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Censor, Y., Gordon, D., Gordon, R.: Component averaging: an efficient iterative parallel algorithm for large and sparse unstructured problems. Parallel Comput. 27(6), 777–808 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Chang, S.Y., Murty, K.G.: The steepest descent gravitational method for linear programming. Discrete Appl. Math. 25(3), 211–239 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Clarke, F.H.: Optimization and Nonsmooth Analysis. Classics in applied mathematics, vol. 2. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1990)Google Scholar
  10. 10.
    Dennis, J.E. Jr., Schnabel, R.B.: Numerical methods for unconstrained optimization and nonlinear equations, volume 16 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, (1996) (Corrected reprint of the 1983 original)Google Scholar
  11. 11.
    Deutsch, F., Hundal, H.: The rate of convergence of Dykstra’s cyclic projections algorithm: the polyhedral case. Numer. Funct. Anal. Optim. 15(5–6), 537–565 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Dykstra, R.L.: An algorithm for restricted least squares regression. J. Am. Statist. Assoc. 78(384), 837–842 (1983)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Ekárt, A., Németh, A.B., Németh, S.Z.: Rapid heuristic projection on simplicial cones (2010) arXiv:1001.1928
  14. 14.
    Foley, J.D., van Dam, A., Feiner, S.K., Hughes, J.F.: Computer Graphics: Principles and Practice. Addison-Wesley systems programming series (1990)Google Scholar
  15. 15.
    Frick, H.: Computing projections into cones generated by a matrix. Biometrical J. 39(8), 975–987 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Hiriart-Urruty, J.-B., Lemaréchal, C.: Convex analysis and minimization algorithms: Fundamentals. I. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], volume 305, Springer, Berlin (1993)Google Scholar
  17. 17.
    Hu, X.: An exact algorithm for projection onto a polyhedral cone. Aust. N. Z. J. Stat. 40(2), 165–170 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Huynh, T., Lassez, C., Lassez, J.-L.: Practical issues on the projection of polyhedral sets. Ann. Math. Artif. Intell. 6(4), 295–315 (1992). Artificial intelligence and mathematics, IICrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Isac, G., Németh, A.B.: Monotonicity of metric projections onto positive cones of ordered Euclidean spaces. Arch. Math. (Basel) 46(6), 568–576 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Isac, G., Németh, A.B.: Isotone projection cones in Euclidean spaces. Ann. Sci. Math. Québec 16(1), 35–52 (1992)zbMATHGoogle Scholar
  21. 21.
    Liu, Z., Fathi, Y.: An active index algorithm for the nearest point problem in a polyhedral cone. Comput. Optim. Appl. 49(3), 435–456 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Liu, Z., Fathi, Y.: The nearest point problem in a polyhedral set and its extensions. Comput. Optim. Appl. 53(1), 115–130 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Mangasarian, O.L.: A generalized Newton method for absolute value equations. Optim. Lett. 3(1), 101–108 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Ming, T., Guo-Liang, T., Hong-Bin, F., Kai Wang, Ng: A fast EM algorithm for quadratic optimization subject to convex constraints. Statist. Sinica 17(3), 945–964 (2007)zbMATHMathSciNetGoogle Scholar
  25. 25.
    Moreau, J.J.: Décomposition orthogonale d’un espace hilbertien selon deux cônes mutuellement polaires. C. R. Acad. Sci. 255, 238–240 (1962)zbMATHMathSciNetGoogle Scholar
  26. 26.
    Morillas, P.M.: Dykstra’s algorithm with strategies for projecting onto certain polyhedral cones. Appl. Math. Comput. 167(1), 635–649 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Murty, K.G.: Sigma Series in Applied Mathematics. Linear complementarity, linear and nonlinear programming. Heldermann, Berlin (1988)Google Scholar
  28. 28.
    Murty, K.G., Fathi, Y.: A critical index algorithm for nearest point problems on simplicial cones. Math. Program. 23(2), 206–215 (1982)CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Németh, A.B., Németh, S.Z.: How to project onto an isotone projection cone. Linear Algebra Appl. 433(1), 41–51 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Németh, S.Z.: Characterization of latticial cones in Hilbert spaces by isotonicity and generalized infimum. Acta Math. Hungar. 127(4), 376–390 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Németh, S.Z.: Isotone retraction cones in Hilbert spaces. Nonlinear Anal. 73(2), 495–499 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Scolnik, H.D., Echebest, N., Guardarucci, M.T., Vacchino, M.C.: Incomplete oblique projections for solving large inconsistent linear systems. Math. Program. 111(1–2, Ser. B), 273–300 (2008)zbMATHMathSciNetGoogle Scholar
  33. 33.
    Shusheng, X.: Estimation of the convergence rate of Dykstra’s cyclic projections algorithm in polyhedral case. Acta Math. Appl. Sinica (English Ser.) 16(2), 217–220 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  34. 34.
    Stewart, G.W.: On the perturbation of pseudo-inverses, projections and linear least squares problems. SIAM Rev. 19(4), 634–662 (1977)CrossRefzbMATHMathSciNetGoogle Scholar
  35. 35.
    Ujvári, M.: On the projection onto a finitely generated cone, 2007, Preprint WP 2007–5. MTA SZTAKI, Laboratory of Operations Research and Decision Systems, Budapest (2007)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Institute of Mathematics and StatisticsGoiâniaBrazil
  2. 2.School of MathematicsThe University of BirminghamBirminghamUK

Personalised recommendations