Research article

General fixed-point method for solving the linear complementarity problem

  • Received: 07 November 2020 Accepted: 25 May 2021 Published: 16 August 2021
  • MSC : 65F10, 65F50

  • In this paper, we consider numerical methods for the linear complementarity problem (LCP). By introducing a positive diagonal parameter matrix, the LCP is transformed into an equivalent fixed-point equation and the equivalence is proved. Based on such equation, the general fixed-point (GFP) method with two cases are proposed and analyzed when the system matrix is a $ P $-matrix. In addition, we provide several concrete sufficient conditions for the proposed method when the system matrix is a symmetric positive definite matrix or an $ H_{+} $-matrix. Meanwhile, we discuss the optimal case for the proposed method. The numerical experiments show that the GFP method is effective and practical.

    Citation: Xi-Ming Fang. General fixed-point method for solving the linear complementarity problem[J]. AIMS Mathematics, 2021, 6(11): 11904-11920. doi: 10.3934/math.2021691

    Related Papers:

  • In this paper, we consider numerical methods for the linear complementarity problem (LCP). By introducing a positive diagonal parameter matrix, the LCP is transformed into an equivalent fixed-point equation and the equivalence is proved. Based on such equation, the general fixed-point (GFP) method with two cases are proposed and analyzed when the system matrix is a $ P $-matrix. In addition, we provide several concrete sufficient conditions for the proposed method when the system matrix is a symmetric positive definite matrix or an $ H_{+} $-matrix. Meanwhile, we discuss the optimal case for the proposed method. The numerical experiments show that the GFP method is effective and practical.



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    [1] S. R. Sacher, On the solution of large, structured linear complementarity problems: The block partitioned case, Appl. Math. Optim., 4 (1977), 347–363. doi: 10.1007/BF01442149
    [2] Z. Z. Bai, Modulus-based matrix splitting iteration methods for linear complementarity problems, Numer. Linear Algebra Appl., 17 (2010), 917–933. doi: 10.1002/nla.680
    [3] C. W. Cryer, The solution of a quadratic programming problem using systematic over-relaxation, SIAM J. Control Optim., 9 (1971), 385–392. doi: 10.1137/0309028
    [4] X. J. Shi, L. Yang, Z. H. Huang, A fixed-point method for the linear complementarity problem arising from American option pricing, Acta Math. Appl. Sinica(Eglish Ser.), 32 (2016), 921–932. doi: 10.1007/s10255-016-0613-6
    [5] W. Li, A general modulus-based matrix splitting method for linear complementarity problems of $H$-matrices, Appl. Math. Lett., 26 (2013), 1159–1164. doi: 10.1016/j.aml.2013.06.015
    [6] C. Zhang, X. J. Chen, Smoothing projected gradient method and its application to stochastic linear complementarity problems, SIAM J. Optim., 20 (2009), 627–649. doi: 10.1137/070702187
    [7] P. H. Calamai, J. J. Mor, Projected gradient methods for linearly constrained problems, Math. Program., 39 (1987), 93–116. doi: 10.1007/BF02592073
    [8] H. Liu, Y. Huang, X. Li, Partial projected Newton method for a class of stochastic linear complementarity problems, Numer. Algorithms, 58 (2011), 593–618. doi: 10.1007/s11075-011-9472-7
    [9] M. D. Koulisianis, T. S. Papatheodorou, Improving projected successive overrelaxation method for linear complementarity problems, Appl. Numer. Math., 45 (2003), 29–40. doi: 10.1016/S0168-9274(02)00233-7
    [10] Y. C. Cheng, On the gradient-projection method for solving the nonsymmetric linear complementarity problem, J. Optimiz. Theory App., 43 (1984), 527–541. doi: 10.1007/BF00935004
    [11] H. Liu, Y. Huang, X. Li, New reformulation and feasible semismooth Newton method for a class of stochastic linear complementarity problems, Appl. Math. Comput., 217 (2011), 9723–9740.
    [12] B. S. He, A projection and contraction method for a class of linear complementarity problems and its application in convex quadratic programming, Appl. Math. Optim., 25 (1992), 247–262. doi: 10.1007/BF01182323
    [13] I. M. Sharaf, A projection algorithm for positive definite linear complementarity problems with applications, Int. J. Manag. Sci. Eng., 12 (2017), 141–147.
    [14] A. Hadjidimos, M. Tzoumas, Nonstationary extrapolated modulus algorithms for the solution of the linear complementarity problem, Linear Algebra Appl., 43 (2009), 197–210.
    [15] 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. doi: 10.1016/j.cam.2016.02.011
    [16] L. L. Zhang, Two-step modulus-based matrix splitting iteration method for linear complementarity problems, Numer. Algorithms, 57 (2011), 83–99. doi: 10.1007/s11075-010-9416-7
    [17] N. Zheng, J. F. Yin, Accelerated modulus-based matrix splitting iteration methods for linear complementarity problem, Numer. Algorithms, 64 (2013), 245–262. doi: 10.1007/s11075-012-9664-9
    [18] S. M. Liu, H. Zheng, W. Li, A general accelerated modulus-based matrix splitting iteration method for solving linear complementarity problems, Calcolo, 53 (2016), 189–199. doi: 10.1007/s10092-015-0143-2
    [19] B. L. Wen, H. Zheng, W. Li, X. F. Peng, The relaxation modulus-based matrix splitting iteration method for solving linear complementarity problems of positive definite matrices, Appl. Math. Comput., 321 (2018), 349–357.
    [20] A. Hadjidimos, M. Lapidakis, M. Tzoumas, On iterative solution for linear complementarity problem with an $H_+$-matrix, SIAM J. Matrix Anal. Appl., 33 (2012), 97–110. doi: 10.1137/100811222
    [21] H. Zheng, S. Vong, On convergence of the modulus-based matrix splitting iteration method for horizontal linear complementarity problems of $H_+$-matrices, Appl. Math. Comput., 369 (2020), 124–890.
    [22] H. Zheng, W. Li, S. W. Vong, A relaxation modulus-based matrix splitting iteration method for solving linear complementarity problems, Numer. Algorithms, 74 (2017), 137–152. doi: 10.1007/s11075-016-0142-7
    [23] H. Zheng, W. Li, W. Qu, A non-modulus linear method for solving the linear complementarity problem, Linear Algebra Appl., 495 (2016), 38–50. doi: 10.1016/j.laa.2016.01.032
    [24] H. Zheng, S. Vong, A two-step modulus-based matrix splitting iteration method for horizontal linear complementarity problems, Numer. Algorithms, 86 (2021), 1791–1810. doi: 10.1007/s11075-020-00954-1
    [25] A. Berman, R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Philadelphia: SIAM Publisher, 1994.
    [26] Z. Z. Bai, On the convergence of the multisplitting methods for the linear complementarity problem, SIAM J. Matrix Anal. Appl., 21 (2006), 67–78.
    [27] G. Csordas, R. S. Varga, Comparisons of regular splittings of matrices, Numer. Math., 44 (1984), 23–35. doi: 10.1007/BF01389752
    [28] A. Frommer, D. B. Szyld, $H$-splittings and two-stage iterative methods, Numer. Math., 63 (1992), 345–356. doi: 10.1007/BF01385865
    [29] K. G. Murty, On the number of solutions to the complementarity problem and spanning properties of complementary cones, Linear Algebra Appl., 5 (1972), 65–108. doi: 10.1016/0024-3795(72)90019-5
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