Skip to main content
SpringerLink
Log in

A class of new modulus-based matrix splitting methods for linear complementarity problem

  • Original Paper
  • Published:
Optimization Letters Aims and scope Submit manuscript

Abstract

In this paper, to economically and fast solve the linear complementarity problem, based on a new equivalent fixed-point form of the linear complementarity problem, we establish a class of new modulus-based matrix splitting methods, which is different from the previously published works. Some sufficient conditions to guarantee the convergence of this new iteration method are presented. Numerical examples are offered to show the efficacy of this new iteration method. Moreover, the comparisons on numerical results show the computational efficiency of this new iteration method advantages over the corresponding modulus method, the modified modulus method and the modulus-based Gauss–Seidel method.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Cottle, R.W., Pang, J.-S., Stone, R.E.: The Linear Complementarity Problem. Academic, San Diego (1992)

    MATH  Google Scholar 

  2. Murty, K.G.: Linear Complementarity. Linear and Nonlinear Programming. Heldermann, Berlin (1988)

    MATH  Google Scholar 

  3. Berman, A., Plemmons, R.J.: Nonnegative Matrices in the Mathematical Sciences. Academic, New York (1979)

    MATH  Google Scholar 

  4. Cottle, R.W., Dantzig, G.B.: Complementary pivot theory of mathematical programming. Linear Algebra Appl. 1, 103–125 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  5. Schäfer, U.: A linear complementarity problem with a P-matrix. SIAM Rev. 46, 189–201 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  6. Zheng, N., Hayami, K., Yin, J.-F.: Modulus-type inner outer iteration methods for nonnegative constrained least squares problems. SIAM J. Matrix Anal. Appl. 37, 1250–1278 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bard, Y.: Nonlinear Parameter Estimarion. Academic Press, New York (1974)

    Google Scholar 

  8. Mangasarian, O.L.: Solutions of symmetric linear complementarity problems by iterative methods. J. Opt. Theory Appl. 22, 465–485 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ahn, B.H.: Solutions of nonsymmetric linear complementarity problems by iterative methods. J. Opt. Theory Appl. 33, 175–185 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  10. Cryer, C.W.: The solution of a quadratic programming using systematic overrelaxation. SIAM J. Control. 9, 385–392 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bai, Z.-Z., Evans, D.J.: Matrix multisplitting relaxation methods for linear complementarity problems. Int. J. Comput. Math. 63, 309–326 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bai, Z.-Z., Evans, D.J.: Chaotic iterative methods for the linear complementarity problems. J. Comput. Appl. Math. 96, 127–138 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  13. Li, S.-G., Jiang, H., Cheng, L.-Z., Liao, X.-K.: IGAOR and multisplitting IGAOR methods for linear complementarity problems. J. Comput. Appl. Math. 235, 2904–2912 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Yuan, D.-J., Song, Y.-Z.: Modified AOR methods for linear complementarity problem. Appl. Math. Comput. 140, 53–67 (2003)

    MathSciNet  MATH  Google Scholar 

  15. Hadjidimos, A., Zhang, L.-L.: Comparison of three classes of algorithms for the solution of the linear complementarity problem with an \(H_{+}\)-matrix. J. Comput. Appl. Math. 336, 175–191 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  16. Noor, M.A.: Fixed point approach for complementarity problems. J. Math. Anal. Appl. 133, 437–448 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  17. Noor, M.A., Zarae, S.: An iterative scheme for complementarity problems. Eng. Anal. 3, 221–224 (1986)

    Article  Google Scholar 

  18. AI-Said, E.A., Noor, M.A.: An iterative scheme for generalized mildly nonlinear complementarity problems. Appl. Math. Lett. 12, 7–11 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  19. AI-Said, E.A., Noor, M.A.: An iterative technique for generalized strongly nonlinear complementarity problems. Appl. Math. Lett. 12, 75–79 (1999)

    MathSciNet  MATH  Google Scholar 

  20. Bai, Z.-Z.: Modulus-based matrix splitting iteration methods for linear complementarity problems. Numer. Linear Algebra Appl. 17, 917–933 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. van Bokhoven, W.M.G.: A Class of Linear Complementarity Problems is Solvable in Polynomial Time. University of Technology, The Netherlands, Unpublished Paper, Dept. of Electrical Engineering (1980)

    Google Scholar 

  22. Kappel, N.W., Watson, L.T.: Iterative algorithms for the linear complementarity problems. Int. J. Comput. Math. 19, 273–297 (1986)

    Article  MATH  Google Scholar 

  23. Dong, J.-L., Jiang, M.-Q.: A modified modulus method for symmetric positive-definite linear complementarity problems. Numer. Linear Algebra Appl. 16, 129–143 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  24. Hadjidimos, A., Tzoumas, M.: Nonstationary extrapolated modulus algorithms for the solution of the linear complementarity problem. Linear Algebra Appl. 431, 197–210 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  25. Bai, Z.-Z., Zhang, L.-L.: Modulus-based synchronous multisplitting iteration methods for linear complementarity problems. Numer. Linear Algebra Appl. 20, 425–439 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  26. Bai, Z.-Z., Zhang, L.-L.: Modulus-based synchronous two-stage multisplitting iteration methods for linear complementarity problems. Numer. Alg. 62, 59–77 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  27. Zheng, N., Yin, J.-F.: Accelerated modulus-based matrix splitting iteration methods for linear complementarity problems. Numer. Alg. 64, 245–262 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  28. Zhang, L.-L.: Two-step modulus-based matrix splitting iteration method for linear complementarity problems. Numer. Alg. 57, 83–99 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  29. Wu, S.-L., Li, C.-X.: Two-sweep modulus-based matrix splitting iteration methods for linear complementarity problems. J. Comput. Math. 302, 327–339 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  30. Li, W.: A general modulus-based matrix splitting method for linear complementarity problems of H-matrices. Appl. Math. Lett. 26, 1159–1164 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  31. Xu, W.-W.: Modified modulus-based matrix splitting iteration methods for linear complementarity problems. Numer. Linear Algebra Appl. 22, 748–760 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  32. Zhang, L.-L., Ren, Z.-R.: Improved convergence theorems of modulus-based matrix splitting iteration methods for linear complementarity problems. Appl. Math. Lett. 26, 638–642 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  33. Hadjidimos, A., Lapidakis, M., Tzoumas, M.: On iterative solution for linear complementarity problem with an \(H_{+}\)-matrix. SIAM J. Matrix Anal. Appl. 33, 97–110 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  34. Ma, C.-F., Huang, N.: Modified modulus-based matrix splitting algorithms for a class of weakly nondifferentiable nonlinear complementarity problems. Appl. Numer. Math. 108, 116–124 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  35. Xia, Z.-C., Li, C.-L.: Modulus-based matrix splitting iteration methods for a class of nonlinear complementarity problem. Appl. Math. Comput. 271, 34–42 (2015)

    MathSciNet  MATH  Google Scholar 

  36. Xie, S.-L., Xu, H.-R., Zeng, J.-P.: Two-step modulus-based matrix splitting iteration method for a class of nonlinear complementarity problems. Linear Algebra Appl. 494, 1–10 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  37. Huang, N., Ma, C.-F.: The modulus-based matrix splitting algorithms for a class of weakly nondifferentiable nonlinear complementarity problems. Numer. Linear Algebra Appl. 23, 558–569 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  38. Hong, J.-T., Li, C.-L.: Modulus-based matrix splitting iteration methods for a class of implicit complementarity problems. Numer. Linear Algebra Appl. 23, 629–641 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  39. Wu, S.-L., Guo, P.: Modulus-based matrix splitting algorithms for the quasi-complementarity problems. Appl. Numer. Math. 132, 127–137 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  40. Mezzadri, F., Galligani, E.: Modulus-based matrix splitting methods for horizontal linear complementarity problems. Numer. Algor. 83, 201–219 (2020)

  41. Bai, Z.-Z., Tong, P.-L.: Iterative methods for linear complementarity problem. J. Univ. Electr. Sci. Tech. China 22, 420–424 (1993). ((In Chinese))

  42. Bai, Z.-Z., Huang, T.-Z.: Accelerated overrelaxation methods for solving linear complementarity problem. J. Univ. Electr. Sci. Tech. China 23, 428–432 (1994). ((In Chinese))

  43. Varga, R.S.: Matrix Iterative Analysis. Prentice-Hall, Englewood Cliffs (1962)

    MATH  Google Scholar 

  44. Frommer, A., Mayer, G.: Convergence of relaxed parallel multisplitting methods. Linear Algebra Appl. 119, 141–152 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  45. Cvetković, L., Hadjidimos, A., Kostić, V.: On the choice of parameters in MAOR type splitting methods for the linear complementarity problem. Numer. Algor. 67, 793–806 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  46. Shi, X.-J., Yang, L., Huang, Z.-H.: A fixed point method for the linear complementarity problem arising from American option pricing. Acta Math. Appl. Sin. 32, 921–932 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  47. Isac, G.: Leray-Uchauder Type Alternatives. Complementarity Problems and Variational Inequalities. Springer, Boston (2006)

    MATH  Google Scholar 

  48. Mezzadri, F.: On the equivalence between some projected and modulus-based splitting methods for linear complementarity problems. Calcolo. 56, 41 (2019)

Download references

Acknowledgements

The authors would like to thank two anonymous referees for providing helpful suggestions, which greatly improved the paper.

Author information

Authors and Affiliations

  1. School of Mathematics, Yunnan Normal University, Kunming, Yunnan, 650500, People’s Republic of China

    Shiliang Wu

  2. School of Mathematics and Computer Science & FJKLMAA, Fujian Normal University, Fuzhou, Fujian, 350117, People’s Republic of China

    Cuixia Li

Authors
  1. Shiliang Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Cuixia Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shiliang Wu.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This research was supported by National Natural Science Foundation of China (No. 11961082)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wu, S., Li, C. A class of new modulus-based matrix splitting methods for linear complementarity problem. Optim Lett 16, 1427–1443 (2022). https://doi.org/10.1007/s11590-021-01781-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-021-01781-6

Keywords

Mathematics Subject Classification

Search

Navigation