Advertisement

Journal of Optimization Theory and Applications

, Volume 180, Issue 2, pp 500–517 | Cite as

Splitting Methods for a Class of Horizontal Linear Complementarity Problems

  • Francesco MezzadriEmail author
  • Emanuele Galligani
Article
  • 38 Downloads

Abstract

In this paper, we propose two splitting methods for solving horizontal linear complementarity problems characterized by matrices with positive diagonal elements. The proposed procedures are based on the Jacobi and on the Gauss–Seidel iterations and differ from existing techniques in that they act directly and simultaneously on both matrices of the problem. We prove the convergence of the methods under some assumptions on the diagonal dominance of the matrices of the problem. Several numerical experiments, including large-scale problems of practical interest, demonstrate the capabilities of the proposed methods in various situations.

Keywords

Horizontal linear complementarity problem Matrix splitting Projected methods 

Mathematics Subject Classification

65K05 65H10 90C33 

Notes

Acknowledgements

The authors desire to thank the anonymous referee for the valuable comments and remarks.

References

  1. 1.
    Cottle, R.W., Pang, J.S., Stone, R.E.: The Linear Complementarity Problem. Classics in Applied Mathematics. SIAM, University City (2009)CrossRefGoogle Scholar
  2. 2.
    Zhang, Y.: On the convergence of a class on infeasible interior-point methods for the horizontal linear complementarity problem. SIAM J. Optimiz. 4(1), 208–227 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Gowda, M.: Reducing a monotone horizontal LCP to an LCP. Appl. Math. Lett. 8(1), 97–100 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Tütüncü, R.H., Todd, M.J.: Reducing horizontal linear complementarity problems. Linear Algebra Appl. 223(224), 717–729 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Ralph, D.: A stable homotopy approach to horizontal linear complementarity problems. Control Cybern. 31, 575–600 (2002)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Gao, X., Wang, J.: Analysis and application of a one-layer neural network for solving horizontal linear complementarity problems. Int. J. Comput. Intell. Syst. 7(4), 724–732 (2014)CrossRefGoogle Scholar
  7. 7.
    Cryer, C.: The solution of a quadratic programming problem using systematic overrelaxation. SIAM J. Control 9, 385–392 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Mangasarian, O.: Solution of symmetric linear complementarity problems by iterative methods. J. Optimiz. Theory Appl. 22, 465–485 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Ahn, B.H.: Solution of nonsymmetric linear complementarity problems by iterative methods. J. Optimiz. Theory App. 33(2), 175–185 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Varga, R.: Matrix Iterative Analysis, 2nd edn. Springer, Berlin (2000)CrossRefzbMATHGoogle Scholar
  11. 11.
    Mezzadri, F., Galligani, E.: An inexact Newton method for solving complementarity problems in hydrodynamic lubrication. Calcolo 55, 1 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Sznajder, R., Gowda, M.S.: Generalizations of \({P}_0\)- and \({P}\)-properties; extended vertical and horizontal linear complementarity problems. Linear Algebra Appl. 223–224, 695–715 (1995)CrossRefzbMATHGoogle Scholar
  13. 13.
    Graham, R., Knuth, D., Patashnik, O.: Concrete Mathematics: A Foundation for Computer Science, 2nd edn. Addison-Wesley, Boston (1994)zbMATHGoogle Scholar
  14. 14.
    Horn, R., Johnson, C.: Matrix Analysis, 2nd edn. Cambridge University Press, Cambridge (2013)zbMATHGoogle Scholar
  15. 15.
    Giacopini, M., Fowell, M., Dini, D., Strozzi, A.: A mass-conserving complementarity formulation to study lubricant films in the presence of cavitation. J. Tribol. 132, 041702 (2010)CrossRefGoogle Scholar
  16. 16.
    Balay, S., Abhyankar, S., Adams, M.F., Brown, J., Brune, P., Buschelman, K., Dalcin, L., Eijkhout, V., Gropp, W.D., Kaushik, D., Knepley, M.G., McInnes, L.C., Rupp, K., Smith, B.F., Zampini, S., Zhang, H., Zhang, H.: PETSc users manual. Tech. Rep. ANL-95/11-Revision 3.8, Argonne National Laboratory (2017). http://www.mcs.anl.gov/petsc

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Engineering “Enzo Ferrari”University of Modena and Reggio EmiliaModenaItaly

Personalised recommendations