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AIMS Mathematics

2024, Volume 9, Issue 6: 14358-14375. doi: 10.3934/math.2024698
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A two-step iteration method for solving vertical nonlinear complementarity problems

  • 1.
    School of Computer Science and Engineering, Macau University of Science and Technology, Macao, China
  • 2.
    School of Mathematics and Statistics, Shaoguan University, Shaoguan, China
  • Received: 21 February 2024 Revised: 06 April 2024 Accepted: 17 April 2024 Published: 19 April 2024

  • MSC : 65F10, 90C33

  • In this paper, for vertical nonlinear complementarity problems, a two-step modulus-based matrix splitting iteration method is established by applying the two-step splitting technique to the modulus-based matrix splitting iteration method. The convergence theorems of the proposed method are given when the number of system matrices is larger than 2. Numerical results show that the convergence rate of the proposed method can be accelerated compared to the existing modulus-based matrix splitting iteration method.

    Citation: Wenxiu Guo, Xiaoping Lu, Hua Zheng. A two-step iteration method for solving vertical nonlinear complementarity problems[J]. AIMS Mathematics, 2024, 9(6): 14358-14375. doi: 10.3934/math.2024698

    Related Papers:

  • In this paper, for vertical nonlinear complementarity problems, a two-step modulus-based matrix splitting iteration method is established by applying the two-step splitting technique to the modulus-based matrix splitting iteration method. The convergence theorems of the proposed method are given when the number of system matrices is larger than 2. Numerical results show that the convergence rate of the proposed method can be accelerated compared to the existing modulus-based matrix splitting iteration method.


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    [1] Z. Bai, On the convergence of the multisplitting methods for the linear complementarity problem, SIAM J. Matrix Anal. Appl., 21 (1999), 67–78. http://dx.doi.org/10.1137/S0895479897324032 doi: 10.1137/S0895479897324032
    [2] Z. Bai, Modulus-based matrix splitting iteration methods for linear complementarity problems, Numer. Linear Algebr., 17 (2010), 917–933. http://dx.doi.org/10.1002/nla.680 doi: 10.1002/nla.680
    [3] M. Bashirizadeh, M. Hajarian, Two-step two-sweep modulus-based matrix splitting iteration method for linear complementarity problems, Numer. Math. Theor. Meth. Appl., 15 (2022), 592–619. http://dx.doi.org/10.4208/nmtma.OA-2021-0131 doi: 10.4208/nmtma.OA-2021-0131
    [4] A. Bensoussan, J. Lions, Applications of variational inequalities in stochastic control, Amsterdam: Elsevier, 1982.
    [5] A. Berman, R. Plemmons, Nonnegative matrix in the mathematical sciences, Philadelphia: SIAM Publisher, 1994. http://dx.doi.org/10.1137/1.9781611971262
    [6] Y. Cao, G. Yang, Q. Shen, Convergence analysis of projected SOR iteration method for a class of vertical linear complementarity problems, Comp. Appl. Math., 42 (2023), 191. http://dx.doi.org/10.1007/s40314-023-02334-6 doi: 10.1007/s40314-023-02334-6
    [7] Y. Cao, A. Wang, Two-step modulus-based matrix splitting iteration methods for implicit complementarity problems, Numer. Algor., 82 (2019), 1377–1394. http://dx.doi.org/10.1007/s11075-019-00660-7 doi: 10.1007/s11075-019-00660-7
    [8] B. Chen, P. Harker, Smooth approximations to nonlinear complementarity problems, SIAM J. Optim., 7 (1997), 403–420. http://dx.doi.org/10.1137/S1052623495280615 doi: 10.1137/S1052623495280615
    [9] R. Cottle, G. Dantzig, A generalization of the linear complementarity problem, J. Comb. Theory, 8 (1970), 79–90. http://dx.doi.org/10.1016/S0021-9800(70)80010-2 doi: 10.1016/S0021-9800(70)80010-2
    [10] R. Cottle, J. Pang, R. Stone, The linear complementarity problem, SanDiego: SIAM Publisher, 1992. http://dx.doi.org/10.1137/1.9780898719000
    [11] A. Ebiefung, M. Kostreva, The generalized Leontief input-output model and its application to the choice of the new technology, Ann. Oper. Res., 44 (1993), 161–172. http://dx.doi.org/10.1007/BF02061065 doi: 10.1007/BF02061065
    [12] X. Fang, General fixed-point method for solving the linear complementarity problem, AIMS Mathematics, 6 (2021), 11904–11920. http://dx.doi.org/10.3934/math.2021691 doi: 10.3934/math.2021691
    [13] X. Fang, The convergence of the modulus-based Jacobi (MJ) iteration method for solving horizontal linear complementarity problems, Comp. Appl. Math., 41 (2022), 134. http://dx.doi.org/10.1007/s40314-022-01842-1 doi: 10.1007/s40314-022-01842-1
    [14] X. Fang, The convergence of modulus-based matrix splitting iteration methods for implicit complementarity problems, J. Comput. Appl. Math., 411 (2022), 114241. http://dx.doi.org/10.1016/j.cam.2022.114241 doi: 10.1016/j.cam.2022.114241
    [15] X. Fang, Z. Gu, Z. Qiao, Convergence of the two-point modulus-based matrix splitting iteration method, J. Appl. Anal. Comput., 13 (2023), 2504–2521. http://dx.doi.org/10.11948/20220400 doi: 10.11948/20220400
    [16] X. Fang, Z. Zhu, The modulus-based matrix double splitting iteration method for linear complementarity problems, Comput. Math. Appl., 78 (2019), 3633–3643. http://dx.doi.org/10.1016/j.camwa.2019.06.012 doi: 10.1016/j.camwa.2019.06.012
    [17] 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
    [18] T. Fujisawa, E. Kuh, Piecewise-linear theory of nonlinear networks, SIAM J. Appl. Math., 22 (1972), 307–328. http://dx.doi.org/10.1137/0122030 doi: 10.1137/0122030
    [19] A. Frommer, D. 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
    [20] M. Seetharama Gowda, R. Sznajder, The generalized order linear complementarity problem, SIAM J. Matrix Anal. Appl., 15 (1994), 779–795. http://dx.doi.org/10.1137/S0895479892237859 doi: 10.1137/S0895479892237859
    [21] W. Guo, H. Zheng, X. Peng, New convergence results of the modulus-based methods for vertical linear complementarity problems, Appl. Math. Lett., 135 (2023), 108444. http://dx.doi.org/10.1016/j.aml.2022.108444 doi: 10.1016/j.aml.2022.108444
    [22] W. Guo, H. Zheng, X. Lu, Y. Zhang, S. Vong, A relaxation two-step parallel modulus method without auxiliary variable for solving large sparse vertical linear complementarity problems, Numer. Algor., in press. http://dx.doi.org/10.1007/s11075-024-01800-4
    [23] J. He, S. Vong, A new kind of modulus-based matrix splitting methods for vertical linear complementarity problems, Appl. Math. Lett., 134 (2022), 108344. http://dx.doi.org/10.1016/j.aml.2022.108344 doi: 10.1016/j.aml.2022.108344
    [24] M. Haddou, M. Tangi, O. J$\acute{e}$r$\acute{e}$my, A generalized direction in interior point method for monotone linear complementarity problems, Optim. Lett., 13 (2019), 35–53. http://dx.doi.org/10.1007/s11590-018-1241-2 doi: 10.1007/s11590-018-1241-2
    [25] J. Hong, C. Li, Modulus-based matrix splitting iteration methods for a class of implicit complementarity problems, Numer. Linear Algebr., 23 (2016), 629–641. http://dx.doi.org/10.1002/nla.2044 doi: 10.1002/nla.2044
    [26] J. Hu, Estimates of $||{B^{-1}C}||_\infty$ and their applications (Chinese), Mathematica Numerica Sinica, 3 (1982), 272–282.
    [27] Y. Ke, C. Ma, H. Zhang, The modulus-based matrix splitting iteration methods for second-order cone linear complementarity problems, Numer. Algor., 79 (2018), 1283–1303. http://dx.doi.org/10.1007/s11075-018-0484-4 doi: 10.1007/s11075-018-0484-4
    [28] Y. Ke, C. Ma, H. Zhang, The relaxation modulus-based matrix splitting iteration methods for circular cone nonlinear complementarity problems, Comp. Appl. Math., 37 (2018), 6795–6820. http://dx.doi.org/10.1007/s40314-018-0687-2 doi: 10.1007/s40314-018-0687-2
    [29] M. Kojima, S. Susumu, H. Shinji, Interior-point methods for the monotone semidefinite linear complementarity problem in symmetric matrices, SIAM J. Optim., 7 (1997), 86–125. http://dx.doi.org/10.1137/S1052623494269035 doi: 10.1137/S1052623494269035
    [30] D. Li, L. Wang, Y. Liu, A relaxation general two-sweep modulus-based matrix splitting iteration method for solving linear complementarity problems, J. Comput. Appl. Math., 409 (2022), 114140. http://dx.doi.org/10.1016/j.cam.2022.114140 doi: 10.1016/j.cam.2022.114140
    [31] F. Mezzadri, A modulus-based formulation for the vertical linear complementarity problems, Numer. Algor., 90 (2022), 1547–1568. http://dx.doi.org/10.1007/s11075-021-01240-4 doi: 10.1007/s11075-021-01240-4
    [32] F. Mezzadri, E. Galligani, Modulus-based matrix splitting methods for horizontal linear complementarity problems, Numer. Algor., 83 (2020), 201–219. http://dx.doi.org/10.1007/s11075-019-00677-y doi: 10.1007/s11075-019-00677-y
    [33] F. Mezzadri, E. Galligani, Projected splitting methods for vertical linear complementarity problems, J. Optim. Theory Appl., 193 (2022), 598–620. http://dx.doi.org/10.1007/s10957-021-01922-y doi: 10.1007/s10957-021-01922-y
    [34] T. Nagae, T. Akamatsu, A generalized complementarity approach to solving real option problems, J. Econ. Dyn. Control., 32 (2008), 1754–1779. http://dx.doi.org/10.1016/j.jedc.2007.04.010 doi: 10.1016/j.jedc.2007.04.010
    [35] K. Oh, The formulation of the mixed lubrication problem as a generalized nonlinear complementarity problem, J. Tribol., 108 (1986), 598–603. http://dx.doi.org/10.1115/1.3261274 doi: 10.1115/1.3261274
    [36] F. Potra, Y. Ye, Interior-point methods for nonlinear complementarity problems, J. Optim. Theory Appl., 88 (1996), 617–642. http://dx.doi.org/10.1007/BF02192201 doi: 10.1007/BF02192201
    [37] H. Qi, L. Liao, Z. Lin, Regularized smoothing approximations to vertical nonlinear complementarity problems, J. Math. Anal. Appl., 230 (1999), 261–276. http://dx.doi.org/10.1006/jmaa.1998.6205 doi: 10.1006/jmaa.1998.6205
    [38] Q. Shi, Q. Shen, T. Tang, A class of two-step modulus-based matrix splitting iteration methods for quasi-complementarity problems, Comp. Appl. Math., 39 (2020), 11. http://dx.doi.org/10.1007/s40314-019-0984-4 doi: 10.1007/s40314-019-0984-4
    [39] M. Sun, Monotonicity of Mangasarian's iterative algorithm for generalized linear complementarity problems, J. Math. Anal. Appl., 144 (1989), 474–485. http://dx.doi.org/10.1016/0022-247X(89)90347-8 doi: 10.1016/0022-247X(89)90347-8
    [40] R. Sznajder, M. Gowda, Generalizations of $P_0$- and $P$-properties; extended vertical and horizontal linear complementarity problems, Linear Algebra Appl., 223-224 (1995), 695–715. http://dx.doi.org/10.1016/0024-3795(93)00184-2 doi: 10.1016/0024-3795(93)00184-2
    [41] Z. Xia, C. Li, Modulus-based matrix splitting iteration methods for a class of nonlinear complementarity problem, Appl. Math. Comput., 271 (2015), 34–42. http://dx.doi.org/10.1016/j.amc.2015.08.108 doi: 10.1016/j.amc.2015.08.108
    [42] S. Xie, Z. Yang, H. Xu, A modulus-based matrix splitting method for the vertical nonlinear complementarity problem, J. Appl. Math. Comput., 69 (2023), 2987–3003. http://dx.doi.org/10.1007/s12190-023-01866-8 doi: 10.1007/s12190-023-01866-8
    [43] S. Xie, H. Xu, J. Zeng, Two-step modulus-based matrix splitting iteration method for a class of nonlinear complementarity problems, Linear Algebra Appl., 494 (2016), 1–10. http://dx.doi.org/10.1016/j.laa.2016.01.002 doi: 10.1016/j.laa.2016.01.002
    [44] L. Zhang, Two-step modulus based matrix splitting iteration for linear complementarity problems, Numer. Algor., 57 (2011), 83–99. http://dx.doi.org/10.1007/s11075-010-9416-7 doi: 10.1007/s11075-010-9416-7
    [45] L. Zhang, Two-step modulus-based synchronous multisplitting iteration methods for linear complementarity problems, J. Comput. Math., 33 (2015), 100–112. http://dx.doi.org/10.4208/jcm.1403-m4195 doi: 10.4208/jcm.1403-m4195
    [46] Y. Zhang, On the convergence of a class of infeasible interior-point methods for the horizontal linear complementarity problem, SIAM J. Optim., 4 (1994), 208–227. http://dx.doi.org/10.1137/0804012 doi: 10.1137/0804012
    [47] Y. Zhang, W. Guo, H. Zheng, S. Vong, A relaxed two-step modulus-based matrix synchronous multisplitting iteration method for linear complementarity problems, Comp. Appl. Math., 43 (2024), 33. http://dx.doi.org/10.1007/s40314-023-02563-9 doi: 10.1007/s40314-023-02563-9
    [48] Y. Zhang, H. Zheng, X. Lu, S. Vong, Modulus-based synchronous multisplitting iteration methods without auxiliary variable for solving vertical linear complementarity problems, Appl. Math. Comput., 458 (2023), 128248. http://dx.doi.org/10.1016/j.amc.2023.128248 doi: 10.1016/j.amc.2023.128248
    [49] Y. Zhang, H. Zheng, X. Lu, S. Vong, A two-step parallel iteration method for large sparse horizontal linear complementarity problems, Appl. Math. Comput., 438 (2023), 127609. http://dx.doi.org/10.1016/j.amc.2022.127609 doi: 10.1016/j.amc.2022.127609
    [50] H. Zheng, X. Lu, S. Vong, A two-step modulus-based matrix splitting iteration method without auxiliary variable for solving vertical linear complementarity problems, Commun. Appl. Math. Comput., in press. http://dx.doi.org/10.1007/s42967-023-00280-y
    [51] H. Zheng, Y. Zhang, X. Lu, S. Vong, Modulus-based synchronous multisplitting iteration methods for large sparse vertical linear complementarity problems, Numer. Algor., 93 (2023), 711–729. http://dx.doi.org/10.1007/s11075-022-01436-2 doi: 10.1007/s11075-022-01436-2
    [52] H. Zheng, S. Vong, A two-step modulus-based matrix splitting iteration method for horizontal linear complementarity problems, Numer. Algor., 86 (2021), 1791–1810. http://dx.doi.org/10.1007/s11075-020-00954-1 doi: 10.1007/s11075-020-00954-1
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