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On the new modulus-based matrix splitting method for linear complementarity problem of \(H_{+}\)-matrix

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Abstract

In this paper, we go on studying the convergence property of the new modulus-based matrix splitting (NMMS) method for linear complementarity problem of \(H_{+}\)-matrix. A new sufficient condition of the NMMS method is obtained, which is weaker than the result in the perviously work by Wu and Li (Opt Lett 16:1427–1443, 2022).

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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. Liao, L.-Z.: A neural network for the linear complementarity problem. Math. Comput. Model. 29, 9–18 (1999)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  5. 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 

  6. Wu, S.-L., Li, C.-X.: A class of new modulus-based matrix splitting methods for linear complementarity problem. Opt. Lett. 16, 1427–1443 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  7. Wu, S.-L., Li, L.: New modulus-based matrix splitting methods for implicit complementarity problem. Numer. Algo. 90, 1735–1754 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  8. Wu, S.-L., Li, C.-X.: A class of modulus-based matrix splitting methods for vertical linear complementarity problem. Optim. https://doi.org/10.1080/02331934.2022.2069021

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

    MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. Hu, J.-G.: Estimates of \(\Vert B^{-1}A\Vert _{\infty }\) and their applications. Math. Numer. Sinica. 4, 272–282 (1982)

    MathSciNet  Google Scholar 

  13. Ahn, B.H.: Iterative methods for linear complementarity problems with upperbounds on primary variables. Math. Program. 26, 295–315 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  14. 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 

  15. 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 

  16. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  20. Van Bokhoven, W.: Piecewise-linear Modelling and Analysis. Eindhoven, Proefschrift (1981)

    Google Scholar 

  21. 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 

  22. 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 

  23. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank two anonymous referees for providing helpful suggestions, which greatly improved the paper. This research was supported by National Natural Science Foundation of China (No. 11961082).

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Correspondence to Shi-Liang Wu.

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Wu, SL. On the new modulus-based matrix splitting method for linear complementarity problem of \(H_{+}\)-matrix. Optim Lett 17, 1669–1678 (2023). https://doi.org/10.1007/s11590-023-01980-3

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