The simplified modulus-based matrix splitting iteration method for the nonlinear complementarity problem

Ximing Fang; Ximing Fang

doi:10.3934/math.2024416

AIMS Mathematics

2024, Volume 9, Issue 4: 8594-8609. doi: 10.3934/math.2024416

Previous Article Next Article

Research article Special Issues

The simplified modulus-based matrix splitting iteration method for the nonlinear complementarity problem

Ximing Fang ^,

School of Mathematics and Statistics, Zhaoqing University, Zhaoqing 526000, China

Received: 14 December 2023 Revised: 27 January 2024 Accepted: 31 January 2024 Published: 29 February 2024
MSC : 65F10, 65N12

In this paper, the simplified modulus-based matrix splitting iteration method was extended to solve the nonlinear complementarity problem, and the convergence conditions were presented from the spectral radius and the matrix norm. Then, for the special cases of this method, we provided some concrete convergence conditions as well as the quasi-optimal parameter matrix. Moreover, some numerical examples were illustrated to show the validity of the convergence results.
- nonlinear complementarity,
- numerical solution,
- iteration method,
- matrix spitting,
- convergence
Citation: Ximing Fang. The simplified modulus-based matrix splitting iteration method for the nonlinear complementarity problem[J]. AIMS Mathematics, 2024, 9(4): 8594-8609. doi: 10.3934/math.2024416

Related Papers:

Abstract

In this paper, the simplified modulus-based matrix splitting iteration method was extended to solve the nonlinear complementarity problem, and the convergence conditions were presented from the spectral radius and the matrix norm. Then, for the special cases of this method, we provided some concrete convergence conditions as well as the quasi-optimal parameter matrix. Moreover, some numerical examples were illustrated to show the validity of the convergence results.

References

[1]	B. N. Pshenichnyi, A. A. Sosnovsky, Nonlinear complementarity problem, Optim., 4 (1992), 355–362. https://doi.org/10.1080/02331939208843832 doi: 10.1080/02331939208843832
[2]	S. Karamardian, The nonlinear complementarity problem with applications, J. Optim. Theory Appl., 4 (1969), 167–181. https://doi.org/10.1007/BF00927414 doi: 10.1007/BF00927414
[3]	Z. C. Xia, C. L. Li, Modulus-based matrix splitting iteration methods for a class of nonlinear complementarity problem, Appl. Math. Comput., 271 (2015), 34–42. https://doi.org/10.1016/j.amc.2015.08.108 doi: 10.1016/j.amc.2015.08.108
[4]	R. Li, J. F. Yin, On the convergence of modulus-based matrix splitting iteration methods for a class of nonlinear complementarity problems with $H+$-matrices, J. Comput. Appl. Math., 342 (2018), 202–209. https://doi.org/10.1016/j.cam.2017.12.029 doi: 10.1016/j.cam.2017.12.029
[5]	J. T. Hong, C. L. Li, Modulus-based matrix splitting iteration methods for a class of nonlinear complementarity problem, Appl. Math. Comput., 23 (2016), 629–641. https://doi.org/10.1002/nla.2044 doi: 10.1002/nla.2044
[6]	N. Huang, C. F. Ma, The modulus-based matrix splitting algorithms for a class of weakly nonlinear complementarity problems, Numer. Linear Algebra., 23 (2016), 558–569. https://doi.org/10.1002/nla.2039 doi: 10.1002/nla.2039
[7]	G. B. Wang, F. P. Tan, Modulus-based multisplitting iteration method for a class of weakly nonlinear complementarity problems, Comput. Appl. Math. Comput., 3 (2021), 419–428. https://doi.org/10.1007/s42967-020-00074-6 doi: 10.1007/s42967-020-00074-6
[8]	X. M. Fang, Convergence of modulus-based matrix splitting iteration method for a class of nonlinear complementarity problems, Numer. Algorithms, 90 (2022), 931–950. https://doi.org/10.1007/s11075-021-01215-5 doi: 10.1007/s11075-021-01215-5
[9]	Z. Z. Bai, Modulus-based matrix splitting iteration methods for linear complementarity problems, Numer. Linear Algebra., 17 (2010), 917–933. https://doi.org/10.1002/nla.680 doi: 10.1002/nla.680
[10]	H. Zheng, S. Vong, A modified modulus-based matrix splitting iteration method for solving implicit complementarity problems, Numer. Algorithms, 82 (2019), 573–592. https://doi.org/10.1007/s11075-018-0614-z doi: 10.1007/s11075-018-0614-z
[11]	F. Mezzadri, E. Galligani, Modulus-based matrix splitting methods for a class of horizontal nonlinear complementarity problems, Numer. Algorithms, 87 (2021), 667–687. https://doi.org/10.1007/s11565-022-00429-2 doi: 10.1007/s11565-022-00429-2
[12]	W. X. Guo, H. Zheng, X. F. Peng, New convergence results of the modulus-based methods for vertical linear complementarity problems, Appl. Math. Lett., 135 (2023), 108444. https://doi.org/10.1016/j.aml.2022.108444 doi: 10.1016/j.aml.2022.108444
[13]	F. Mezzadri, E. Galligani, Modulus-based matrix splitting methods for horizontal linear complementarity problems, Numer. Algorithms, 83 (2020), 201–219. https://doi.org/10.1007/s11075-019-00677-y doi: 10.1007/s11075-019-00677-y
[14]	F. Mezzadri, A modulus-based formulation for the vertical linear complementarity problems, Numer. Algorithms, 90 (2022), 1547–1568. https://doi.org/10.1007/S11075-021-01240-4 doi: 10.1007/S11075-021-01240-4
[15]	H. Zheng, Y. X. Zhang, X. P. Lu, S. Vong, Modulus-based synchronous multisplitting iteration methods for large sparse vertical linear complementarity problems, Numer. Algorithms, 93 (2023), 711–729. https://doi.org/10.1007/s11075-022-01436-2 doi: 10.1007/s11075-022-01436-2
[16]	X. M. Fang, The convergence of a modulus-based matrix splitting iteration method for solving the implicit complementarity problems, J. Appl. Math. Comput., 69 (2023), 853–870. https://doi.org/10.1016/j.cam.2022.114241 doi: 10.1016/j.cam.2022.114241
[17]	J. Long, S. Zeng, A projection-filter method for solving nonlinear complementarity problems, Appl. Math. Comput., 216 (2010), 300–307. https://doi.org/10.1016/j.amc.2010.01.063 doi: 10.1016/j.amc.2010.01.063
[18]	A. Bnouhachem, M. A. Noor, An interior proximal point algorithm for nonlinear complementarity problems, Nonlinear Anal. Hybri., 4 (2010), 371–380. https://doi.org/10.1016/j.nahs.2009.09.010 doi: 10.1016/j.nahs.2009.09.010
[19]	Y. Qin, Z. Yu, A smoothing least square method for nonlinear complementarity problem, Math. Method. Appl. Sci., 36 (2013), 1783–1789. https://doi.org/10.1002/mma.2724 doi: 10.1002/mma.2724
[20]	A. Hadjidimos, M. Lapidakis, M. Tzoumas, On iterative solution for linear complementarity problem with an $H_+$-matrix, SIAM J. Matrix Anal. A, 33 (2012), 97–110. https://doi.org/10.1137/100811222 doi: 10.1137/100811222
[21]	L. Cvetkovic, A. Hadjidimos, V. Kostic, On the choice of parameters in MAOR type splitting methods for the linear complementarity problem, Numer. Algorithms, 4 (2014), 793–806. https://doi.org/10.1007/s11075-014-9824-1 doi: 10.1007/s11075-014-9824-1
[22]	S. L. Wu, C. X. Li, A class of new modulus-based matrix splitting methods for linear complementarity problem, Optim. Lett., 5 (2022), 1427–1443. https://doi.org/10.1007/s11590-021-01781-6 doi: 10.1007/s11590-021-01781-6
[23]	W. Li, A general modulus-based matrix splitting method for linear complementarity problems of $H$-matrices, Appl. Math. Lett., 26 (2013), 1159–1164. https://doi.org/10.1016/j.aml.2013.06.015 doi: 10.1016/j.aml.2013.06.015
[24]	N. Zheng, J. F. Yin, Accelerated modulus-based matrix splitting iteration methods for linear complementarity problem, Numer. Algorithms, 64 (2013), 245–262. https://doi.org/10.1007/s11075-012-9664-9 doi: 10.1007/s11075-012-9664-9
[25]	L. L. Zhang, Two-stage multisplitting iteration methods using modulus-based matrix splitting as inner iteration for Linear complementarity problems, J. Optimiz. Theory Appl., 160 (2014), 189–203. https://doi.org/10.1007/s10957-013-0362-0 doi: 10.1007/s10957-013-0362-0
[26]	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. https://doi.org/10.1016/j.cam.2016.02.011 doi: 10.1016/j.cam.2016.02.011
[27]	L. Jia, X. Wang, A generalized two-step modulus-based matrix splitting iteration method for implicit complementarity problems of $H+$-matrices, Filomat, 33 (2019), 4875–4888. https://doi.org/10.2298/fil1915875j doi: 10.2298/fil1915875j
[28]	H. Zheng, S. Vong, A two-step modulus-based matrix splitting iteration method for horizontal linear complementarity problems, Numer. Algorithms, 86 (2021), 1791–1810. https://doi.org/10.1007/s11075-020-00954-1 doi: 10.1007/s11075-020-00954-1
[29]	G. Csordas, R. S. Varga, Comparisons of regular splittings of matrices, Numer. Math., 44 (1984), 23–35. https://doi.org/10.1007/BF01389752 doi: 10.1007/BF01389752
[30]	A. Berman, R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, New York: Academic Press, 1994. https://doi.org/10.1137/1023089

Reader Comments

Your name:*

Email:*
© 2024 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)