Search Advanced

AIMS Mathematics

2023, Volume 8, Issue 6: 13368-13389. doi: 10.3934/math.2023677
Previous Article Next Article
Research article

A new relaxed acceleration two-sweep modulus-based matrix splitting iteration method for solving linear complementarity problems

  • 1.
    Institute for Optimization and Decision Analytics, Liaoning Technical University, Fuxin 123000, China
  • 2.
    School of Mathematics and Statistics, FJKLMAA and Center for Applied Mathematics of Fujian Province, Fujian Normal University, Fuzhou 350007, China
  • 3.
    LMIB of the Ministry of Education, School of Mathematical Sciences, Beihang University, Beijing 100191, China
  • Received: 26 January 2023 Revised: 13 March 2023 Accepted: 15 March 2023 Published: 06 April 2023

  • MSC : 65F10, 65H10, 90C30

  • A new relaxed acceleration two-sweep modulus-based matrix splitting (NRATMMS) iteration method is developed to solve linear complementarity problems. The convergence of the NRATMMS method is established with the system matrix $ A $ being an $ H_{+} $-matrix. Numerical experiments show that the proposed method is superior to some existing algorithms under appropriate conditions.

    Citation: Dongmei Yu, Yiming Zhang, Cairong Chen, Deren Han. A new relaxed acceleration two-sweep modulus-based matrix splitting iteration method for solving linear complementarity problems[J]. AIMS Mathematics, 2023, 8(6): 13368-13389. doi: 10.3934/math.2023677

    Related Papers:

  • A new relaxed acceleration two-sweep modulus-based matrix splitting (NRATMMS) iteration method is developed to solve linear complementarity problems. The convergence of the NRATMMS method is established with the system matrix $ A $ being an $ H_{+} $-matrix. Numerical experiments show that the proposed method is superior to some existing algorithms under appropriate conditions.


    加载中


    [1] R. W. Cottle, J. S. Pang, R. E. Stone, The linear complementarity problem, San Diego: Academic Press, 1992. http://doi.org/10.1137/1.9780898719000
    [2] K. G. Murty, Linear complementarity, linear and nonlinear programming, Berlin: Heldermann, 1988.
    [3] U. Schäfer, A linear complementarity problem with a $P$-matrix, SIAM Rev., 46 (2004), 189–201. http://dx.doi.org/10.1137/S0036144502420083 doi: 10.1137/S0036144502420083
    [4] N. W. Kappel, L. T. Watson, Iterative algorithms for the linear complementarity problem, Int. J. Comput. Math., 19 (1986), 273–297. http://dx.doi.org/10.1080/00207168608803522 doi: 10.1080/00207168608803522
    [5] Z. Z. Bai, On the monotone convergence of the projected iteration methods for linear complementarity problem, Numer. Math. J. Chinese Univ. (English Ser.), 5 (1996), 228–233.
    [6] Y. Li, P. Dai, Generalized AOR methods for linear complementarity problem, Appl. Math. Comput., 188 (2007), 7–18. http://dx.doi.org/10.1016/j.amc.2006.09.067 doi: 10.1016/j.amc.2006.09.067
    [7] O. L. Mangasarian, Solution of symmetric linear complementarity problems by iterative methods, J. Optim. Theory Appl., 22 (1977), 465–485. http://dx.doi.org/10.1007/BF01268170 doi: 10.1007/BF01268170
    [8] H. S. Najafi, S. A. Edalatpanah, On the convergence regions of generalized accelerated overrelaxation method for linear complementarity problems, J. Optim. Theory Appl., 156 (2013), 859–866. http://dx.doi.org/10.1007/s10957-012-0135-1 doi: 10.1007/s10957-012-0135-1
    [9] 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://dx.doi.org/10.1137/S0895479897324032 doi: 10.1137/S0895479897324032
    [10] Z. Z. Bai, Parallel chaotic multisplitting iterative methods for the large sparse linear complementarity problem, J. Comput. Math., 19 (2001), 281–292.
    [11] 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://dx.doi.org/10.1002/nla.1835 doi: 10.1002/nla.1835
    [12] 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://dx.doi.org/10.1080/00207160211927 doi: 10.1080/00207160211927
    [13] Z. Z. Bai, L. L. Zhang, Modulus-based synchronous two-stage multisplitting iteration methods for linear complementarity problems, Numer. Algorithms, 62 (2013), 59–77. https://dx.doi.org/10.1007/s11075-012-9566-x doi: 10.1007/s11075-012-9566-x
    [14] N. Machida, M. Fukushima, T. Ibaraki, A multisplitting method for symmetric linear complementarity problems, J. Comput. Appl. Math., 62 (1995), 217–227. http://dx.doi.org/10.1016/0377-0427(94)00103-2 doi: 10.1016/0377-0427(94)00103-2
    [15] C. Kanzow, Inexact semismooth Newton methods for large-scale complementarity problems, Optim. Methods Softw., 19 (2004), 309–325. http://dx.doi.org/10.1080/10556780310001636369 doi: 10.1080/10556780310001636369
    [16] Z. Sun, J. P. Zeng, A monotone semismooth Newton type method for a class of complementarity problems, J. Comput. Appl. Math., 235 (2011), 1261–1274. http://dx.doi.org/10.1016/j.cam.2010.08.012 doi: 10.1016/j.cam.2010.08.012
    [17] Z. Z. Bai, L. L. Zhang, Modulus-based multigrid methods for linear complementarity problems, Numer. Linear Algebra Appl., 24 (2017), e2105. https://dx.doi.org/10.1002/nla.2105 doi: 10.1002/nla.2105
    [18] L. Jia, X. Wang, X. Y. Xiao, The nonlinear lopsided HSS-like modulus-based matrix splitting iteration method for linear complementarity problems with positive-definite matrices, Commun. Appl. Math. Comput., 3 (2021), 109–122. http://dx.doi.org/10.1007/s42967-019-00038-5 doi: 10.1007/s42967-019-00038-5
    [19] L. L. Zhang, Two-stage multisplitting iteration methods using modulus-based matrix splitting as inner iteration for linear complementarity problems, J. Optim. Theory Appl., 160 (2014), 189–203. http://dx.doi.org/10.1007/s10957-013-0362-0 doi: 10.1007/s10957-013-0362-0
    [20] Z. Z. Bai, Modulus-based matrix splitting iteration methods for linear complementarity problems, Numer. Linear Algebra Appl., 17 (2010), 917–933. https://dx.doi.org/10.1002/nla.680 doi: 10.1002/nla.680
    [21] A. Hadjidimos, M. Tzoumas, Nonstationary extrapolated modulus algorithms for the solution of the linear complementarity problem, Linear Algebra Appl., 431 (2009), 197–210. http://dx.doi.org/10.1016/j.laa.2009.02.024 doi: 10.1016/j.laa.2009.02.024
    [22] 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. http://dx.doi.org/10.1002/nla.609 doi: 10.1002/nla.609
    [23] N. Zheng, J. F. Yin, Accelerated modulus-based matrix splitting iteration methods for linear complementarity problem, Numer. Algorithms, 64 (2013), 245–262. http://dx.doi.org/10.1007/s11075-012-9664-9 doi: 10.1007/s11075-012-9664-9
    [24] H. Zheng, W. Li, S. Vong, A relaxation modulus-based matrix splitting iteration method for solving linear complementarity problems, Numer. Algorithms, 74 (2016), 137–152. http://dx.doi.org/10.1007/s11075-016-0142-7 doi: 10.1007/s11075-016-0142-7
    [25] W. Li, A general modulus-based matrix splitting method for linear complementarity problems of $H$-matrices, Appl. Math. Lett., 26 (2013), 1159–1164. http://dx.doi.org/10.1016/j.aml.2013.06.015 doi: 10.1016/j.aml.2013.06.015
    [26] S. Q. Shen, T. Z. Huang, Convergence and comparison theorems for double splittings of matrices, Comput. Math. Appl., 51 (2006), 1751–1760. http://dx.doi.org/10.1016/j.camwa.2006.02.006 doi: 10.1016/j.camwa.2006.02.006
    [27] Z. I. Woźnicki, Estimation of the optimum relaxation factors in partial factorization iterative methods, SIAM J. Matrix Anal. Appl., 14 (1993), 59–73. http://dx.doi.org/10.1137/0614005 doi: 10.1137/0614005
    [28] S. L. Wu, C. X. Li, Two-sweep modulus-based matrix splitting iteration methods for linear complementarity problems, J. Comput. Appl. Math., 302 (2016), 327–339. http://dx.doi.org/10.1016/j.cam.2016.02.011 doi: 10.1016/j.cam.2016.02.011
    [29] H. Ren, X. Wang, X. B. Tang, T. Wang, The general two-sweep modulus-based matrix splitting iteration method for solving linear complementarity problems, Comput. Math. Appl., 77 (2019), 1071–1081. http://dx.doi.org/10.1016/j.camwa.2018.10.040 doi: 10.1016/j.camwa.2018.10.040
    [30] X. Peng, M. Wang, W. Li, A relaxation two-sweep modulus-based matrix splitting iteration method for linear complementarity problems, East Asian J. Appl. Math., 9 (2019), 102–121. http://dx.doi.org/10.4208/eajam.020318.220618 doi: 10.4208/eajam.020318.220618
    [31] Z. G. Huang, J. J. Cui, Accelerated relaxation modulus-based matrix splitting iteration method for linear complementarity problems, Bull. Malays. Math. Sci. Soc., 44 (2021), 2175–2213. http://dx.doi.org/10.1007/s40840-020-01049-9 doi: 10.1007/s40840-020-01049-9
    [32] P. F. Dai, J. Li, J. Bai, J. Qiu, A preconditioned two-step modulus-based matrix splitting iteration method for linear complementarity problem, Appl. Math. Comput., 348 (2019), 542–551. http://dx.doi.org/10.1016/j.amc.2018.12.012 doi: 10.1016/j.amc.2018.12.012
    [33] B. Huang, C. Ma, Accelerated modulus-based matrix splitting iteration method for a class of nonlinear complementarity problems, Comput. Appl. Math., 37 (2018), 3053–3076. http://dx.doi.org/10.1007/s40314-017-0496-z doi: 10.1007/s40314-017-0496-z
    [34] N. Li, J. Ding, J. F. Yin, Modified relaxation two-sweep modulus-based matrix splitting iteration method for solving a class of implicit complementarity problems, J. Comput. Appl. Math., 413 (2022), 114370. http://dx.doi.org/10.1016/j.cam.2022.114370 doi: 10.1016/j.cam.2022.114370
    [35] S. Liu, H. Zheng, W. Li, A general accelerated modulus-based matrix splitting iteration method for solving linear complementarity problems, Calcolo, 53 (2015), 189–199. http://dx.doi.org/10.1007/s10092-015-0143-2 doi: 10.1007/s10092-015-0143-2
    [36] F. Mezzadri, On the equivalence between some projected and modulus-based splitting methods for linear complementarity problems, Calcolo, 56 (2019), 41. http://dx.doi.org/10.1007/s10092-019-0337-0 doi: 10.1007/s10092-019-0337-0
    [37] H. Ren, X. Wang, X. B. Tang, T. Wang, A preconditioned general two-step modulus-based matrix splitting iteration method for linear complementarity problems of $H_{+}$-matrices, Numer. Algorithms, 82 (2019), 969–986. http://dx.doi.org/10.1007/s11075-018-0637-5 doi: 10.1007/s11075-018-0637-5
    [38] Y. J. Wu, W. H. Zhang, A. L. Yang, Modulus-based inexact non-alternating preconditioned splitting method for linear complementarity problems, Linear Multilinear Algebra, in press. http://dx.doi.org/10.1080/03081087.2021.1991874
    [39] L. L. Zhang, Two-step modulus-based matrix splitting iteration method for linear complementarity problems, Numer. Algorithms, 57 (2011), 83–99. http://dx.doi.org/10.1007/s11075-010-9416-7 doi: 10.1007/s11075-010-9416-7
    [40] L. L. Zhang, Z. R. Ren, Improved convergence theorems of modulus-based matrix splitting iteration methods for linear complementarity problems, Appl. Math. Lett., 26 (2013), 638–642. http://dx.doi.org/10.1016/j.aml.2013.01.001 doi: 10.1016/j.aml.2013.01.001
    [41] N. Zheng, J. F. Yin, Convergence of accelerated modulus-based matrix splitting iteration methods for linear complementarity problem with an $H_+$-matrix, J. Comput. Appl. Math., 260 (2014), 281–293. http://dx.doi.org/10.1016/j.cam.2013.09.079 doi: 10.1016/j.cam.2013.09.079
    [42] Z. Z. Bai, P. L. Tong, Iterative methods for linear complementarity problem, J. UEST China, 22 (1993), 420–424.
    [43] Z. Z. Bai, T. Z. Huang, Accelerated overrelaxation methods for solving linear complementarity problem, J. UEST China, 23 (1994), 428–432.
    [44] S. L. Wu, C. X. Li, A class of new modulus-based matrix splitting methods for linear complementarity problem, Optim. Lett., 16 (2022), 1427–1443. http://dx.doi.org/10.1007/s11590-021-01781-6 doi: 10.1007/s11590-021-01781-6
    [45] A. Frommer, D. B. Szyld, H-splittings and two-stage iterative methods, Numer. Math., 63 (1992), 345–356. http://dx.doi.org/10.1007/BF01385865 doi: 10.1007/BF01385865
    [46] Z. Z. Bai, D. J. Evans, Matrix multisplitting relaxation methods for linear complementarity problems, Int. J. Comput. Math., 63 (1997), 309–326. https://dx.doi.org/10.1080/00207169708804569 doi: 10.1080/00207169708804569
    [47] J. G. Hu, Estimates of $\|B^{-1}A\|_{\infty}$ and their applications, Math. Numer. Sin., 3 (1982), 272–282.
    [48] A. Frommer, G. Mayer, Convergence of relaxed parallel multisplitting methods, Linear Algebra Appl., 119 (1989), 141–152. http://dx.doi.org/10.1016/0024-3795(89)90074-8 doi: 10.1016/0024-3795(89)90074-8
    [49] A. Berman, R. J. Plemmons, Nonnegative matrices in the mathematical sciences, Philadelphia: SIAM, 1994. http://dx.doi.org/10.1137/1.9781611971262
    [50] X. J. Shi, Y. Lei, Z. H. Huang, A fixed point method for the linear complementarity problem arising from american option pricing, Acta Math. Appl. Sin. Engl. Ser., 32 (2016), 921–932. http://dx.doi.org/10.1007/s10255-016-0613-6 doi: 10.1007/s10255-016-0613-6
  • Reader Comments
  • © 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索
AIMS Mathematics
1.8 3.4

Metrics

Article views(1188) PDF downloads(78) Cited by(2)

Preview PDF
Download XML
Export Citation
Article outline

Figures and Tables

Tables(3)

Other Articles By Authors

Related pages

Tools

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog