Journal of Optimization Theory and Applications

, Volume 162, Issue 3, pp 931–953 | Cite as

The Bound-Constrained Conjugate Gradient Method for Non-negative Matrices



Existing conjugate gradient (CG)-based methods for convex quadratic programs with bound constraints require many iterations for solving elastic contact problems. These algorithms are too cautious in expanding the active set and are hampered by frequent restarting of the CG iteration. We propose a new algorithm called the Bound-Constrained Conjugate Gradient method (BCCG). It combines the CG method with an active-set strategy, which truncates variables crossing their bounds and continues (using the Polak–Ribière formula) instead of restarting CG. We provide a case with n=3 that demonstrates that this method may fail on general cases, but we conjecture that it always works if the system matrix A is non-negative. Numerical results demonstrate the effectiveness of the method for large-scale elastic contact problems.


Quadratic program Bound constraints Conjugate gradients method Active set algorithm Elastic contact problem 


  1. 1.
    Kalker, J.: Three-Dimensional Elastic Bodies in Rolling Contact. Solid Mechanics and Its Applications. Kluwer Academic, Amsterdam (1990) CrossRefGoogle Scholar
  2. 2.
    Vollebregt, E.: User guide for CONTACT, Vollebregt & Kalker’s rolling and sliding contact model. Tech. Rep. TR09-03. version 13.1, VORtech BV (2013). See
  3. 3.
    Hager, W., Zhang, H.: A new active set algorithm for box constrained optimization. SIAM J. Optim. 17(2), 526–557 (2006) MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Moré, J., Toraldo, G.: On the solution of large quadratic programming problems with bound constraints. SIAM J. Optim. 1, 93–113 (1991) MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Bertsekas, D.: Nonlinear Programming. Athena Scientific, Belmont (1999) MATHGoogle Scholar
  6. 6.
    Nocedal, J., Wright, S.: Numerical Optimization. Springer, New York (2000) Google Scholar
  7. 7.
    Hestenes, M., Stiefel, E.: Methods of conjugate gradients for solving linear systems. J. Res. Natl. Bur. Stand. 49(6), 409–436 (1952) MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Shewchuk, J.: An introduction to the conjugate gradient method without the agonizing pain. Tech. rep., Carnegie mellon University, Pittsburgh, PA, USA (1994) Google Scholar
  9. 9.
    Barrett, R., Berry, M., Chan, T., Demmel, J., Donato, J., Dongarra, J., Eijkhout, V., Pozo, R., Romine, C., van der Vorst, H.: Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods. SIAM, Philadelphia (1994) Google Scholar
  10. 10.
    Saad, Y.: Iterative Methods for Sparse Linear Systems, 2nd edn. SIAM, Philadelphia (2003) MATHCrossRefGoogle Scholar
  11. 11.
    Hestenes, M.: Conjugate Direction Methods in Optimization, Applications of Mathematics vol. 12. Springer, New York (1980) CrossRefGoogle Scholar
  12. 12.
    Polyak, B.: The conjugate gradient method in extremal problems. USSR Comput. Math. Math. Phys. 9, 94–112 (1969) CrossRefGoogle Scholar
  13. 13.
    O’Leary, D.: A generalized conjugate gradient algorithm for solving a class of quadratic programming problems. Linear Algebra Appl. 34, 371–399 (1980) MATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Dembo, R., Tulowitzki, U.: On the minimization of quadratic functions subject to box constraints. Tech. Rep., School of Organization and Management, Yale University, New Haven, CT (1983) Google Scholar
  15. 15.
    Yang, E., Tolle, J.: A class of methods for solving large convex quadratic programs subject to box constraints. Math. Program. 51, 223–228 (1991) MATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Birgin, E., Martínez, J.: Large-scale active-set box-constrained optimization method with spectral projected gradients. Comput. Optim. Appl. 22, 101–125 (2002) CrossRefGoogle Scholar
  17. 17.
    Dostál, Z., Schöberl, J.: Minimizing quadratic functions subject to bound constraints with the rate of convergence and finite termination. Comput. Optim. Appl. 30, 23–43 (2005) MATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Polonsky, I., Keer, L.: A numerical method for solving rough contact problems based on the multi-level multi-summation and conjugate gradients techniques. Wear 231, 206–219 (1999) CrossRefGoogle Scholar
  19. 19.
    Sainsot, P., Lubrecht, A.: Efficient solution of the dry contact for rough surfaces: a comparison of fast Fourier transform and multigrid methods. Proc. Inst. Mech. Eng., Part J J. Eng. Tribol. 225, 441–448 (2011) Google Scholar
  20. 20.
    Polak, E., Ribière, G.: Note sur la convergence de méthodes de directions conjugées. Rev. Francaise Inform. Rech. Opér. 16, 35–43 (1969) Google Scholar
  21. 21.
    Press, W., Teukolsky, S., Vetterling, W., Flannery, B.: Numerical Recipes in Fortran 77 The Art of Scientific Computing 2nd edn. Cambridge University Press, Cambridge (1992) Google Scholar
  22. 22.
    Vollebregt, E.: A Gauss–Seidel type solver for special convex programs, with application to frictional contact mechanics. J. Optim. Theory Appl. 87(1), 47–67 (1995) MATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    HinterMüller, M., Ito, K., Kunisch, K.: The primal-dual active set strategy as a semismooth Newton method. SIAM J. Optim. 13, 865–888 (2003) MATHCrossRefGoogle Scholar
  24. 24.
    Meyer, C.: Matrix Analysis and Applied Linear Algebra. SIAM, Philadelphia (2000) MATHCrossRefGoogle Scholar
  25. 25.
    Vollebregt, E.: A new solver for the elastic normal contact problem using conjugate gradients, deflation, and an FFT-based preconditioner. J. Comput. Phys., Part A 257, 333–351 (2014). doi:10.1016/ MathSciNetCrossRefGoogle Scholar
  26. 26.
    Nogi, T., Kato, T.: Influence of a hard surface layer on the limit of elastic contact—Part I: Analysis using a real surface model. ASME J. Tribol. 119, 493–500 (1997) CrossRefGoogle Scholar
  27. 27.
    Webster, M., Sayles, R.: A numerical model for the elastic contact of real rough surfaces. ASME J. Tribol. 108, 314–320 (1986) CrossRefGoogle Scholar
  28. 28.
    Johnson, K.: Contact Mechanics. Cambridge University Press, Cambridge (1985) MATHCrossRefGoogle Scholar
  29. 29.
    Dekking, F., Kalker, J., Vollebregt, E.: Simulation of rough, elastic contacts. J. Appl. Mech. 64, 361–368 (1997) MATHCrossRefGoogle Scholar
  30. 30.
    Vollebregt, E.: Condition and eigenvalues of the elastic normal contact problem. Report EV/M13.003, VORtech BV (2013). Available upon request Google Scholar
  31. 31.
    Liu, S., Wang, Q.: Studying contact stress fields caused by surface tractions with a discrete convolution and Fast Fourier transform algorithm. ASME J. Tribol. 124, 36–45 (2002) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.DIAMDelft University of TechnologyDelftThe Netherlands
  2. 2.VORtech BVDelftThe Netherlands

Personalised recommendations