CONVERGENCE OF THE TWO-POINT MODULUS-BASED MATRIX SPLITTING ITERATION METHOD

Ximing Fang; Ze Gu; Zhijun Qiao

doi:10.11948/20220400

2023 Volume 13 Issue 5

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Article navigation > Journal of Applied Analysis & Computation > 2023 > 13(5): 2504-2521

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Ximing Fang, Ze Gu, Zhijun Qiao. CONVERGENCE OF THE TWO-POINT MODULUS-BASED MATRIX SPLITTING ITERATION METHOD[J]. Journal of Applied Analysis & Computation, 2023, 13(5): 2504-2521. doi: 10.11948/20220400

Citation:

Ximing Fang, Ze Gu, Zhijun Qiao. CONVERGENCE OF THE TWO-POINT MODULUS-BASED MATRIX SPLITTING ITERATION METHOD[J]. Journal of Applied Analysis & Computation, 2023, 13(5): 2504-2521. doi: 10.11948/20220400

CONVERGENCE OF THE TWO-POINT MODULUS-BASED MATRIX SPLITTING ITERATION METHOD

1.
School of Mathematics and Statistics, Zhaoqing University, 526000 Zhaoqing, China
2.
School of Mathematical and Statistical Sciences, The University of Texas Rio Grande Valley, 78539 Edinburg, USA

Author Bio: Email: fangxm504@163.com(X. M. Fang); Email: guze528@sina.com(Z. Gu)
Corresponding author: Email: zhijun.qiao@utrgv.edu(Z. J. Qiao)
Fund Project: The authors were supported by the Natural Science Foundation of Guangdong Province (No. 2023A1515011911), the Guangdong Basic and Applied Basic Research Foundation (No. 2022A1515011081), the Characteristic Innovation Project of Department of Education of Guangdong Province (No. 2020KTSCX159), the Innovative Research Team Project of Zhaoqing Uiversity and the Scientific Research Ability Enhancement Program for Excellent Young Teachers of Zhaoqing University, the Technology Innovation Guidance Project of Zhaoqing (No. 2022040315016) and the University of Texas System President's Endowed Professorship Program (No. 450000123)

Abstract

Abstract

In this paper, we discuss the convergence of the two-point modulus-based matrix splitting iteration method for solving the linear complementarity problem. Some convergence conditions are presented from the spectral radius and the matrix norm when the system matrix is a $ P $-matrix. Besides, the quasi-optimal cases of the method are studied. Numerical experiments are provided to show the presented results.
- Linear complementarity problem /
- matrix splitting /
- convergence
MSC: 65F10, 65F50

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References

[1]	B. H. Ahn, Solution of nonsymmetric linear complementarity problems by iterative methods, J. Optim. Theory Appl., 1981, 33(2), 175–185. doi: 10.1007/BF00935545 CrossRef Google Scholar
[2]	W. M. G. Bokhoven van, Piecewise-linear modelling and analysis, Technische Hogeschool Eindhoven, Eindhoven, 1981. Google Scholar
[3]	Z. Bai, Modulus-based matrix splitting iteration methods for linear complementarity problems, Numer. Linear Algebra Appl., 2010, 17(6), 917–933. doi: 10.1002/nla.680 CrossRef Google Scholar
[4]	L. Cvetkovic, A. Hadjidimos and V. Kostic, On the choice of parameters in MAOR type splitting methods for the linear complementarity problem, Numer. Algorithms, 2014, 4(67), 793–806. Google Scholar
[5]	R. W. Cottle, J. Pang and R. E. Stone, The Linear Complementarity Problem, Academic Press, Inc., Bostion, MA, 1992. Google Scholar
[6]	C. W. Cryer, The solution of a quadratic programming problem using systematic over-relaxation, SIAM J. Control Optim., 1971, 9(3), 385–392. doi: 10.1137/0309028 CrossRef Google Scholar
[7]	J. Dong and M. Jiang, A modified modulus method for symmetric positive-definite linear complementarity problems, Numer. Linear Algebra Appl., 2009, 16(2), 129–143. doi: 10.1002/nla.609 CrossRef Google Scholar
[8]	P. Dai, J. Li, J. Bai and J. Qiu, A preconditioned two-step modulus-based matrix splitting iteration method for linear complementarity problem, Appl. Math. Comput., 2019, 348, 542–551. Google Scholar
[9]	A. Frommer and G. Mayer, Convergence of relaxed parallel multisplitting methods, Linear Algebra Appl., 1989, 119, 141–152. doi: 10.1016/0024-3795(89)90074-8 CrossRef Google Scholar
[10]	A. Frommer and D. B. Szyld, H-splittings and two-stage iterative methods, Numer. Math., 1992, 63(1), 345–356. doi: 10.1007/BF01385865 CrossRef Google Scholar
[11]	M. C. Ferris and J. Pang, Complementarity and Variational Problems, State of the Art, SIAM, Philadephia, Pennsylvania, 1997. Google Scholar
[12]	A. Hadjidimos and M. Tzoumas, Nonstationary extrapolated modulus algorithms for the solution of the linear complementarity problem, Linear Algebra Appl., 2009, 431(1–2), 197–210. doi: 10.1016/j.laa.2009.02.024 CrossRef Google Scholar
[13]	A. Hadjidimos, M. Lapidakis and M. Tzoumas, On iterative solution for linear complementarity problem with an H₊-matrix, SIAM J. Matrix Anal. Appl., 2012, 33(1), 97–110. doi: 10.1137/100811222 CrossRef Google Scholar
[14]	M. Kojima, N. Megiddo and Y. Ye, An interior point potential reduction algorithm for the linear complementarity problem, Math. Program., 1992, 54(1–3), 267–279. doi: 10.1007/BF01586054 CrossRef Google Scholar
[15]	S. Liu, H. Zheng and W. Li, A general accelerated modulus-based matrix splitting iteration method for solving linear complementarity problems, Calcolo, 2016, 53(2), 189–199. doi: 10.1007/s10092-015-0143-2 CrossRef Google Scholar
[16]	W. Li, A general modulus-based matrix splitting method for linear complementarity problems of H-matrices, Appl. Math. Lett., 2013, 26(12), 1159–1164. doi: 10.1016/j.aml.2013.06.015 CrossRef Google Scholar
[17]	I. Marek and D. B. Szyld, Comparison theorems for weak splittings of bounded operators, Numer. Math., 1990, 58(1), 387–397. doi: 10.1007/BF01385632 CrossRef Google Scholar
[18]	K. G. Murty, Linear Complementarity, Linear and Nonlinear Programming, Heldermann Verlag, Berlin, 1988. Google Scholar
[19]	H. Ren, X. Wang, X. Tang and T. Wang, The general two-sweep modulus-based matrix splitting iteration method for solving linear complementarity problems, Comput. Math. Appl., 2019, 77, 1071–1081. doi: 10.1016/j.camwa.2018.10.040 CrossRef Google Scholar
[20]	X. Shi, L. Yang and Z. Huang, A fixed point method for the linear complementarity problem arising from American option pricing, Acta Math. Appl. Sin., 2016, 32(4), 921–932. doi: 10.1007/s10255-016-0613-6 CrossRef Google Scholar
[21]	S. Wu and C. Li, Two-sweep modulus-based matrix splitting iteration methods for linear complementarity problems, J. Comput. Appl. Math., 2016, 302, 327–339. doi: 10.1016/j.cam.2016.02.011 CrossRef Google Scholar
[22]	S. Wu and C. Li, A generalized Newton method for non-Hermitian positive definite linear complementarity problem, Calcolo, 2017, 54(1), 43–56. doi: 10.1007/s10092-016-0175-2 CrossRef Google Scholar
[23]	X. Wu, X. Peng and W. Li, A preconditioned general modulus-based matrix splitting iteration method for linear complementarity problems of H-matrices, Numer. Algorithms, 2018, 79(4), 1131–1146. doi: 10.1007/s11075-018-0477-3 CrossRef Google Scholar
[24]	H. Zheng and W. Li, The modulus-based nonsmooth Newton¡¯s method for solving linear complementarity problems, J. Comput. Appl. Math., 2015, 288, 116–126. doi: 10.1016/j.cam.2015.04.006 CrossRef Google Scholar
[25]	H. Zheng, W. Li and S. Vong, A relaxation modulus-based matrix splitting iteration method for solving linear complementarity problems, Numer. Algorithms, 2017, 74, 137–152. doi: 10.1007/s11075-016-0142-7 CrossRef Google Scholar
[26]	L. Zhang, Two-step modulus-based matrix splitting iteration method for linear complementarity problems, Numer. Algorithms, 2011, 57(1), 83–99. doi: 10.1007/s11075-010-9416-7 CrossRef Google Scholar
[27]	L. Zhang, Two-stage multisplitting iteration methods using modulus-based matrix splitting as inner iteration for linear complementarity problems, J. Optim. Theory Appl., 2014, 160(1), 189–203. doi: 10.1007/s10957-013-0362-0 CrossRef Google Scholar
[28]	N. Zheng and J. Yin, Accelerated modulus-based matrix splitting iteration methods for linear complementarity problem, Numer. Algorithms, 2013, 64(2), 245–262. doi: 10.1007/s11075-012-9664-9 CrossRef Google Scholar