Advertisement

Mathematical Programming

, Volume 47, Issue 1–3, pp 37–51 | Cite as

On the convergence properties of Hildreth's quadratic programming algorithm

  • Alfredo N. Iusem
  • Alvaro R. De Pierro
Article

Abstract

We prove the linear convergence rate of Hildreth's method for quadratic programming, in both its sequential and simulateneous versions. We give bounds on the asymptotic error constant and compare these bounds to those given by Mandel for the cyclic relaxation method for solving linear inequalities.

Key words

Quadratic programming almost cyclic relaxation bounds of convergence 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    S. Agmon, “The relaxation method for linear inequalities,”Canadian Journal of Mathematics 6 (1954) 382–392.Google Scholar
  2. [2]
    Y. Censor, “Row-action methods for huge and sparse systems and their applications,”SIAM Review 23 (1981) 444–466.Google Scholar
  3. [3]
    Y. Censor and G.T. Herman, “Row generation methods for feasibility and optimization problems involving sparse matrices and their applications,” in: I.S. Duff and G.W. Stewart, eds.,Sparse Matrix Proceedings 1978 (Society for Industrial and Applied Mathematics, Philadelphia, PA, 1979) pp. 197–219.Google Scholar
  4. [4]
    J.L. Goffin, “The relaxation method for solving systems of linear inequalities,”Mathematics of Operations Research 5 (1980) 388–414.Google Scholar
  5. [5]
    R. Gordon and G.T. Herman, “Three-dimensional reconstruction from projections: A review of algorithms,”International Review of Cytology 38 (1974) 111–151.Google Scholar
  6. [6]
    G.T. Herman and A. Lent, “Iterative reconstruction algorithms,”Computers in Biology and Medicine 6 (1976) 273–294.Google Scholar
  7. [7]
    G.T. Herman and A. Lent, “A family of iterative quadratic optimization algorithms for pairs of inequalities with applications in diagnostic radiology,”Mathematical Programming Study 9 (1978) 15–29.Google Scholar
  8. [8]
    C. Hildreth, “A quadratic programming procedure,”Naval Research Logistics Quarterly 4 (1957) 79–85. [Erratum,ibid., p. 361.]Google Scholar
  9. [9]
    A.N. Iusem and A.R. De Pierro, “A simultaneous iterative method for computing projections on polyhedra,”SIAM Journal on Control 25 (1986) 231–243.Google Scholar
  10. [10]
    S. Kaczmarz, “Angenäherte Auflösung von Systemen linearer Gleichungen,”Bulletin International de l'Academie Polonaise de Sciences et Lettres A35 (1937) 355–357.Google Scholar
  11. [11]
    A. Lent and Y. Censor, “Extensions of Hildreth's row-action method for quadratic programming,”SIAM Journal on Control 18 (1980) 444–454.Google Scholar
  12. [12]
    J. Mandel, “Convergence of the cyclical relaxation method for linear inequalities,”Mathematical Programming 30 (1984) 218–228.Google Scholar
  13. [13]
    O.L. Mangasarian, “Solution of symmetric linear complementarity problems by iterative methods,”Journal of Optimization Theory and Applications 22 (1977) 465–485.Google Scholar
  14. [14]
    T.H. Motzkin and I.J. Schoenberg, “The relaxation method for linear inequalities,”Canadian Journal of Mathematics 6 (1954) 393–404.Google Scholar
  15. [15]
    J.S. Pang, “More results on the convergence of iterative methods for the symmetric linear complementarity problem,”Journal of Optimization Theory and Applications 49 (1986) 107–134.Google Scholar

Copyright information

© The Mathematical Programming Society, Inc. 1990

Authors and Affiliations

  • Alfredo N. Iusem
    • 1
  • Alvaro R. De Pierro
    • 2
  1. 1.Instituto de Matemática Pura e AplicadaJardim BotânicoRio de JaneiroBrazil
  2. 2.Medical Image Processing Group, Department of RadiologyHospital of the University of PennsylvaniaPhiladelphiaUSA

Personalised recommendations