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AIMS Mathematics

2022, Volume 7, Issue 6: 10994-11014. doi: 10.3934/math.2022614
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Research article

A general modulus-based matrix splitting method for quasi-complementarity problem

  • 1.
    School of Information Science and Technology, Nantong University, Nantong 226019, China
  • 2.
    School of Transportation and Civil Engineering, Nantong University, Nantong 226019, China
  • 3.
    School of Sciences, Nantong University, Nantong 226019, China
  • Received: 22 December 2021 Revised: 14 March 2022 Accepted: 28 March 2022 Published: 06 April 2022

  • MSC : 65F10

  • For large sparse quasi-complementarity problem (QCP), Wu and Guo [35] recently studied a modulus-based matrix splitting (MMS) iteration method, which belongs to a class of inner-outer iteration methods. In order to improve the convergence rate of the inner iteration so as to get fast convergence rate of the outer iteration, a general MMS (GMMS) iteration method is proposed in this paper. Convergence analyses on the GMMS method are studied in detail when the system matrix is either an $ H_{+} $-matrix or a positive definite matrix. In the case of $ H_{+} $-matrix, weaker convergence condition of the GMMS iteration method is obtained. Finally, two numerical experiments are conducted and the results indicate that the new proposed GMMS method achieves a better performance than the MMS iteration method.

    Citation: Chen-Can Zhou, Qin-Qin Shen, Geng-Chen Yang, Quan Shi. A general modulus-based matrix splitting method for quasi-complementarity problem[J]. AIMS Mathematics, 2022, 7(6): 10994-11014. doi: 10.3934/math.2022614

    Related Papers:

  • For large sparse quasi-complementarity problem (QCP), Wu and Guo [35] recently studied a modulus-based matrix splitting (MMS) iteration method, which belongs to a class of inner-outer iteration methods. In order to improve the convergence rate of the inner iteration so as to get fast convergence rate of the outer iteration, a general MMS (GMMS) iteration method is proposed in this paper. Convergence analyses on the GMMS method are studied in detail when the system matrix is either an $ H_{+} $-matrix or a positive definite matrix. In the case of $ H_{+} $-matrix, weaker convergence condition of the GMMS iteration method is obtained. Finally, two numerical experiments are conducted and the results indicate that the new proposed GMMS method achieves a better performance than the MMS iteration method.


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    [1] Z. Z. Bai, On the convergence of the multisplitting methods for the linear complementarity problem, SIAM J. Matrix Anal. Appl., 21 (1999), 67–78. https://doi.org/10.1137/S0895479897324032 doi: 10.1137/S0895479897324032
    [2] Z. Z. Bai, Modulus-based matrix splitting iteration methods for linear complementarity problems, Numer. Linear Algebra Appl., 17 (2010), 917–933. https://doi.org/10.1002/nla.680 doi: 10.1002/nla.680
    [3] Z. Z. Bai, A class of two-stage iterative methods for systems of weakly nonlinear equations, Numer. Algor., 14 (1997), 295–319. https://doi.org/10.1023/A:1019125332723 doi: 10.1023/A:1019125332723
    [4] Z. Z. Bai, D. J. Evans, Matrix multisplitting relaxation methods for linear complementarity problems, Int. J. Comput. Math., 63 (1997), 309–326. https://doi.org/10.1080/00207169708804569 doi: 10.1080/00207169708804569
    [5] Z. Z. Bai, D. J. Evans, Matrix multisplitting methods with applications to linear complementarity problems: Parallel asynchronous methods, Int. J. Comput. Math., 79 (2002), 205–232. https://doi.org/10.1080/00207160211927 doi: 10.1080/00207160211927
    [6] Z. Z. Bai, J. Y. Pan, Matrix analysis and computations, Philadelphia: SIAM, 2021.
    [7] Z. Z. Bai, X. Yang, On HSS-based iteration methods for weakly nonlinear systems, Appl. Numer. Math., 59 (2009), 2923–2936. https://doi.org/10.1016/j.apnum.2009.06.005 doi: 10.1016/j.apnum.2009.06.005
    [8] Z. Z. Bai, L. L. Zhang, Modulus-based synchronous multisplitting iteration methods for linear complementarity problems, Numer. Linear Algebra Appl., 20 (2013), 425–439. https://doi.org/10.1002/nla.1835 doi: 10.1002/nla.1835
    [9] A. Berman, R. J. Plemmons, Nonnegative matrices in the mathematical sciences, Philadelphia: SIAM, 1994.
    [10] Y. Cao, Q. Shi, S. L. Zhu, A relaxed generalized Newton iteration method for generalized absolute value equations, AIMS Math., 6 (2021), 1258–1275. https://doi.org/10.3934/math.2021078 doi: 10.3934/math.2021078
    [11] Y. Cao, A. Wang, Two-step modulus-based matrix splitting iteration methods for implicit complementarity problems, Numer. Algor., 82 (2019), 1377–1394. https://doi.org/10.1007/s11075-019-00660-7 doi: 10.1007/s11075-019-00660-7
    [12] R. W. Cottle, F. Giannessi, J. L. Lions, Variational inequalities and complementarity problems: Theory and applications, Hoboken, New Jersey: John Wiley & Sons, 1980.
    [13] R. W. Cottle, J. S. Pang, R. E. Stone, The linear complementarity problem, Philadelphia: SIAM, 2009.
    [14] J. L. Dong, M. Q. Jiang, A modified modulus method for symmetric positive-definite linear complementarity problems, Numer. Linear Algebra Appl., 16 (2009), 129–143. https://doi.org/10.1002/nla.609 doi: 10.1002/nla.609
    [15] J. L. Dong, J. B. Gao, F. J. Ju, J. H. Shen, Modulus methods for nonnegatively constrained image restoration, SIAM J. Imaging Sci., 9 (2016), 1226–1246. https://doi.org/10.1137/15M1045892 doi: 10.1137/15M1045892
    [16] M. C. Ferris, J. S. Pang, Engineering and economic applications of complementarity problems, SIAM Rev., 39 (1997), 669–713. https://doi.org/10.1137/S0036144595285963 doi: 10.1137/S0036144595285963
    [17] A. Frommer, G. Mayer, Convergence of relaxed parallel multisplitting methods, Linear Algebra Appl., 119 (1989), 141–152. https://doi.org/10.1016/0024-3795(89)90074-8 doi: 10.1016/0024-3795(89)90074-8
    [18] A. Frommer, D. B. Szyld, $H$-splittings and two-stage iterative methods, Numer. Math., 63 (1992), 345–356. https://doi.org/10.1007/BF01385865 doi: 10.1007/BF01385865
    [19] J. T. Hong, C. L. Li, Modulus-based matrix splitting iteration methods for a class of implicit complementarity problems, Numer. Linear Algebra Appl., 23 (2016), 629–641. https://doi.org/10.1002/nla.2044 doi: 10.1002/nla.2044
    [20] J. G. Hu, Estimates of $\|B^{-1}A\|_{\infty}$ and their applications (in Chinese), Math. Numer. Sin., 3 (1982), 272–282.
    [21] G. Isac, Complementarity problems, Berlin, Heidelberg: Springer, 1992. https://doi.org/10.1007/BFb0084653
    [22] L. Jia, X. Wang, A generalized two-step modulus-based matrix splitting iteration method for implicit complementarity problems of $H_{+}$-matrices, Filomat, 33 (2019), 4875–4888. https://doi.org/10.2298/FIL1915875J doi: 10.2298/FIL1915875J
    [23] Y. F. Ke, C. F. Ma, On the convergence analysis of two-step modulus-based matrix splitting iteration method for linear complementarity problems, Appl. Math. Comput., 243 (2014), 413–418. https://doi.org/10.1016/j.amc.2014.05.119 doi: 10.1016/j.amc.2014.05.119
    [24] M. D. Koulisianis, T. S. Papatheodorou, Improving projected successive overrelaxation method for linear complementarity problems, Appl. Numer. Math., 45 (2003), 29–40. https://doi.org/10.1016/S0168-9274(02)00233-7 doi: 10.1016/S0168-9274(02)00233-7
    [25] R. Li, J. F. Yin, Accelerated modulus-based matrix splitting iteration methods for a restricted class of nonlinear complementarity problems, Numer. Algor., 75 (2017), 339–358. https://doi.org/10.1007/s11075-016-0243-3 doi: 10.1007/s11075-016-0243-3
    [26] W. Li, A general modulus-based matrix splitting method for linear complementarity problems of $H$-matrices, Appl. Math. Lett., 26 (2013), 1159–1164. https://doi.org/10.1016/j.aml.2013.06.015 doi: 10.1016/j.aml.2013.06.015
    [27] F. Mezzadri, E. Galligani, An inexact Newton method for solving complementarity problems in hydrodynamic lubrication, Calcolo, 55 (2018), 1–28. https://doi.org/10.1007/s10092-018-0244-9 doi: 10.1007/s10092-018-0244-9
    [28] M. A. Noor, The quasi-complementarity problem, J. Math. Anal. Appl., 130 (1988), 344–353. https://doi.org/10.1016/0022-247X(88)90310-1 doi: 10.1016/0022-247X(88)90310-1
    [29] J. S. Pang, Inexact Newton methods for the nonlinear complementarity problem, Math. Program., 36 (1986), 54–71. https://doi.org/10.1007/BF02591989 doi: 10.1007/BF02591989
    [30] G. Ramadurai, S. V. Ukkusuri, J. Y. Zhao, J. S. Pang, Linear complementarity formulation for single bottleneck model with heterogeneous commuters, Transport. Res. B Meth., 44 (2010), 193–214. https://doi.org/10.1016/j.trb.2009.07.005 doi: 10.1016/j.trb.2009.07.005
    [31] U. Sch{ä}fer, On the modulus algorithm for the linear complementarity problem, Oper. Res. Lett., 32 (2004), 350–354. https://doi.org/10.1016/j.orl.2003.11.004 doi: 10.1016/j.orl.2003.11.004
    [32] Q. Shi, Q. Q. Shen, T. P. Tang, A class of two-step modulus-based matrix splitting iteration methods for quasi-complementarity problems, Comput. Appl. Math., 39 (2020), 1–23. https://doi.org/10.1007/s40314-019-0984-4 doi: 10.1007/s40314-019-0984-4
    [33] R. S. Varga, Matrix iterative analysis, Berlin, Heidelberg: Springer, 2000.
    [34] A. Wang, Y. Cao, J. X. Chen, Modified Newton-type iteration methods for generalized absolute value equations, J. Optim. Theory Appl., 181 (2019), 216–230. https://doi.org/10.1007/s10957-018-1439-6 doi: 10.1007/s10957-018-1439-6
    [35] S. L. Wu, P. Guo, Modulus-based matrix splitting algorithms for the quasi-complementarity problems, Appl. Numer. Math., 132 (2018), 127–137. https://doi.org/10.1016/j.apnum.2018.05.017 doi: 10.1016/j.apnum.2018.05.017
    [36] L. L. Zhang, Two-step modulus-based matrix splitting iteration method for linear complementarity problems, Numer. Algor., 57 (2011), 83–99. https://doi.org/10.1007/s11075-010-9416-7 doi: 10.1007/s11075-010-9416-7
    [37] L. L. Zhang, Z. R. Ren, A modified modulus-based multigrid method for linear complementarity problems arising from free boundary problems, Appl. Numer. Math., 164 (2021), 89–100. https://doi.org/10.1016/j.apnum.2020.09.008 doi: 10.1016/j.apnum.2020.09.008
    [38] H. Zheng, L. Liu, A two-step modulus-based matrix splitting iteration method for solving nonlinear complementarity problems of $H_{+}$-matrices, Comput. Appl. Math., 37 (2018), 5410–5423. https://doi.org/10.1007/s40314-018-0646-y doi: 10.1007/s40314-018-0646-y
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