JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2025, Vol. 60 ›› Issue (12): 55-65.doi: 10.6040/j.issn.1671-9352.0.2024.213

Previous Articles     Next Articles

Modulus-based matrix splitting iteration method for the implicit complementarity problem

WEN Shuhong1, KE Yifen2*, LI Keliang2   

  1. 1. Department of Data Science and Statistics, Fuzhou University Zhicheng College, Fuzhou 350002, Fujian, China;
    2. School of Mathematics and Statistics, Fujian Normal University, Fuzhou 350117, Fujian, China
  • Published:2025-12-10

PDF (PC)

33

Abstract: A new modulus-based matrix splitting iteration method is mainly established for a class of implicit complementarity problem in this paper. Any specificity of the initial vector is not depended on by the proposed method, and only one linear system needs to be solved at each iteration step. In particular, when the system matrix is a H+-matrix, the sufficient conditions for global convergence are given. Numerical experiments are presented to show the efficiency of the proposed method,which is superior to the modified modulus-based matrix splitting iteration method in terms of numerical performances.

Key words: implicit complementarity problem, modulus-based method, convergence

CLC Number: 

  • O241
[1] COTTLE R W, PANG J S, STONE R E. The linear complementarity problem[M]. Boston: Academic Press, 1992.
[2] FERRI M C, PANG J S. Engineering and economic applications of complementarity problems[J]. SIAM Rev, 1997, 39:669-713.
[3] MURTY K G. Linear complementarity, linear and nonlinear programming[M]. Berlin: Heldermann Verlag, 1988.
[4] BAI Z Z. Modulus-based matrix splitting iteration methods for linear complementarity problems[J]. Numer Linear Algebra Appl, 2010, 6:917-933.
[5] ZHANG L L. Two-step modulus-based matrix splitting iteration method for linear complementarity problems[J]. Numer Algorithms, 2011, 57(1):83-99.
[6] KE Y F, MA C F. On the convergence analysis of two-step modulus-based matrix splitting iteration method for linear complementarity problems[J]. Appl Math Comput, 2014, 243:413-418.
[7] BAI Z Z, ZHANG L L. Modulus-based synchronous multisplitting iteration methods for linear complementarity problems[J]. Numer Linear Algebra Appl, 2013, 20(3):425-439.
[8] BAI Z Z, ZHANG L L. Modulus-based synchronous two-stage multisplitting iteration methods for linear complementarity problems[J]. Numer Algorithms, 2013, 62:59-77.
[9] ZHENG N, YIN J F. Accelerated modulus-based matrix splitting iteration methods for linear complementarity problem[J]. Numer Algorithms, 2013, 64(2):245-262.
[10] ZHANG Y X, ZHENG H, LU X P, et al. A two-step parallel iteration method for large sparse horizontal linear complementarity problems[J]. Appl Math Comput, 2023, 438:127609.
[11] ZHENG H, VONG S. On the modulus-based successive overrelaxation iteration method for horizontal linear complementarity problems arising from hydrodynamic lubrication[J]. Appl Math Comput, 2021, 402:126165.
[12] ZHENG H, VONG S. On convergence of the modulus-based matrix splitting iteration method for horizontal linear complementarity problems of H+-matrices[J]. Appl Math Comput, 2020, 369:124890.
[13] ZHENG H, VONG S, LIU L. A direct preconditioned modulus-based iteration method for solving nonlinear complementarity problems of H-matrices[J]. Appl Math Comput, 2019, 353:396-405.
[14] ZHENG H, LI W, VONG S. A relaxation modulus-based matrix splitting iteration method for solving linear complementarity problems[J]. Numer Algorithms, 2017, 74:137-152.
[15] 郑华,罗静. 一类H矩阵线性互补问题的预处理二步模基矩阵分裂迭代方法[J]. 计算数学,2018,40(1):24-31. ZHENG Hua, LUO Jing. A preconditioned two-steps modulus-based matrix splitting iteration method for solving linear complementarity problems of H-matrices[J]. Mathematica Numerica Sinica, 2018, 40(1):24-31.
[16] 彭小飞. 线性互补问题的广义松弛两步模基矩阵分裂迭代法[J]. 华南师范大学学报(自然科学版),2019,51(4):93-99. PENG Xiaofei. A general relaxation two-sweep modulus-based matrix splitting iteration method for linear complementarity problems[J]. Journal of South China Normal University(Natural Science Edition),2019, 51(4): 93-99.
[17] HONG J T, LI C L. Modulus-based matrix splitting iteration methods for a class of implicit complementarity problems[J]. Numer Linear Algebra Appl, 2016, 23:629-641.
[18] LI C L, HONG J T. Modulus-based synchronous multisplitting iteration methods for an implicit complementarity problem[J]. E Asian J Appl Math, 2017, 7(2):363-375.
[19] ZHENG H, VONG S. A modified modulus-based matrix splitting iteration method for solving implicit complementarity problems[J]. Numer Algorithms, 2019, 82:573-592.
[20] PANG J S. On the convergence of a basic iterative method for the implicit complementarity problems[J]. Optimiz Theory Appl, 1982, 37:149-162.
[21] CAO Y, WANG A. Two-step modulus-based matrix splitting iteration methods for implicit complementarity problems[J]. Numer Algorithms, 2019, 82:1377-1394.
[22] LI N, DING J, YIN J F. Modified relaxation two-sweep modulus-based matrix splitting iteration method for solving a class of implicit complementarity problems[J]. Comput Appl Math, 2022, 413:114370.
[23] WU S L, LI L. New modulus-based matrix splitting methods for implicit complementarity problem[J]. Numer Algorithms, 2022, 90(4):1735-1754.
[24] LI Z Z, ZHANG H, OU-YANG L. The selection of the optimal parameter in the modulus-based matrix splitting algorithm for linear complementarity problems[J]. Comput Optim Appl, 2021, 80(2):617-638.
[25] LI Z Z, ZHANG H. On estimation of the optimal parameter of the modulus-based matrix splitting algorithm for linear complementarity problems on second-order cones[J]. Numer Linear Algebra Appl, 2023, 30(4):e2480.
[26] LI Z Z, ZHANG H, OU-YANG L. Anderson accelerating the preconditioned modulus approach for linear complementarity problems on second-order cones[J]. Numer Algorithms, 2022, 91(2):803-839.
[27] LI Z Z, ZHANG H. Anderson acceleration of the modulus-based matrix splitting algorithms for horizontal nonlinear complementarity systems[J]. Numer Linear Algebra Appl, 2022, 29(5):e2438.
[28] 黎科良,柯艺芬,马昌凤. 求解一类隐式互补问题的加速模系矩阵分裂迭代方法[J]. 应用数学,2023,36(4):1025-1033. LI Keliang, KE Yifen, MA Changfeng. Accelerated modulus-based matrix splitting iteration method for solving a class of implicit complementarity problems[J]. Mathematica Applicata,2023, 36(4):1025-1033.
[29] FROMMER A, MAYER G. Convergence of relaxed parallel multisplitting methods[J]. Linear Algebra Appl, 1989, 119:141-152.
[30] BERMAN A, PLEMMONS R J. Nonnegative matrix in the mathematical sciences[M]. Philadelphia: SIAM Publisher, 1994.
[1] ZHOU Yulan, WEI Wanying, LIU Cuicui, YANG Qingqing. Properties of power-number operators in the functional space of discrete time normal martingale [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2025, 60(2): 85-95.
[2] Hongwu QIN,Lizheng WANG,Yu FU,Muxuan SUI,Binggao HE. Grey wolf optimization algorithm based on multi-strategy combination and its application [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2024, 59(3): 51-60.
[3] Yang WANG. Block triangular splitting and its preconditioning iterative algorithms for a class of complex symmetric linear systems [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2024, 59(10): 1-9.
[4] Xiufeng YANG,Jianghua FAN. Nonemptiness and boundedness of weakly efficient solution sets for vector mixed variational inequalities [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2023, 58(7): 121-126.
[5] Huimin ZHU,Zinan WANG,Min GAO,Wenzhi YANG. The consistency of CUSUM-type estimator in variance change-point model [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2023, 58(7): 106-114.
[6] CHEN Jing-jing, YANG Yan-tao. Modified inertial projection algorithm for solving variational inequality and fixed point problems [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2023, 58(3): 64-76.
[7] MENG Xu-dong. Stability and extended well-posedness of the solution sets for set optimization problems [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2022, 57(2): 98-110.
[8] JIA Cheng-cheng, WU Qun-ying. Complete convergence of weighted sums for END sequence under sub-linear expectation space [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2022, 57(10): 79-87.
[9] WANG Miao-miao, DING Xiao-li, LI Jia-min. Waveform relaxation methods for fractional stochastic delay differential equations [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2022, 57(1): 101-110.
[10] WANG Song-hua, LUO Dan, LI Yong. A class of newly modified WYL conjugate gradient algorithms [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2021, 56(9): 87-95.
[11] WANG Zeng-zhen, LIU Hua-yong, ZHA Dong-dong. Progressive-iterative approximation by the triangular β-B curves with shape parameter [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2021, 56(6): 81-94.
[12] DENG Xiao-qin, WU Qun-ying. Precise asymptotics of complete moment convergence for ρ-mixing sequence [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2020, 55(6): 32-40.
[13] CHEN Qian-zhu, HU Hai-ping. Optimizing the efficiency of first-order methods for the distance to an optimal solution of smooth convex functions [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2020, 55(4): 102-107.
[14] ZHANG Ya-jing, TAO Hui-ling, LI Xiang, SHEN Ai-ting. Complete moment convergence and complete integral convergence for END random variables [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2020, 55(1): 5-11.
[15] ZHANG Ru, HAN Xu, LIU Xiao-gang. Convergence and contractivity of boundary value methods for nonlinear delay differential equations [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(8): 97-101.
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
[1] XIE Juan-ying1, 2, ZHANG Yan1, XIE Wei-xin2, 3, GAO Xin-bo2. A novel rough K-means clustering algorithm based on the weight of density[J]. J4, 2010, 45(7): 1 -6 .
[2] WU Zhi-jun,SHEN Dan-dan. Architecture and key technologies of network-enabled next generation global flight tracking based on information integration and sharing[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2016, 51(11): 1 -6 .
[3] QU Shouning, FU Aifang, LI Jing, LIU Jing. Forecasting stock market risks based on the flexible neural tree[J]. J4, 2009, 44(11): 44 -47 .
[4] REN Min1,2, ZHANG Guang-hui1. Absorbing probabilities of random walks in an independent random  environment convergence in distribution on the half-line[J]. J4, 2013, 48(1): 93 -99 .
[5] ZHANG Xin-dong ,WANG Qiu-hua . An improvement on the 2-order-derivative-free iterative method of Newton[J]. J4, 2007, 42(7): 72 -76 .
[6] ZHAO Jun1, ZHAO Jing2, FAN Ting-jun1*, YUAN Wen-peng1,3, ZHANG Zheng1, CONG Ri-shan1. Purification and anti-tumor activity examination of water-soluble asterosaponin from Asterias rollestoni Bell[J]. J4, 2013, 48(1): 30 -35 .
[7] . Study of catalysts for synthesis of Schiff bases under microwaveassisted[J]. J4, 2009, 44(7): 18 -21 .
[8] CAO Huiren 1, YANG Yongjie 2, DU Yi 1, WANG Dengyu 1, KONG Weihua 1. Screening of differentially expressed genes in heatstressed mouse testis
by construction of a subtractive cDNA library[J]. J4, 2009, 44(3): 22 -27 .
[9] WANG Hong-mei,XIAO Min*,LI Zheng-yi,LI Yu-mei,QIAN Xin-min, . Screening and identification of βgalactosidaseproducing microorganism and enzymatic synthesis of galactooligosaccharides using its transgalactosylation[J]. J4, 2006, 41(1): 133 -139 .
[10] ZHANG Jia-li, MIAO Lian-ying, SONG Wen-yao. Edge colorings of 1-planar graphs for maximum degree eight #br# without adjacent 4-cycles[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2014, 49(04): 18 -23 .
Copyright 2018 © Editorial office of JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)
Supported by Beijing Magtech Co.Ltd