THE MODULUS-BASED MATRIX SPLITTING METHOD WITH INNER ITERATION FOR A CLASS OF NONLINEAR COMPLEMENTARITY PROBLEMS

Changfeng Ma; Ting Wang

doi:10.11948/20210515

2023 Volume 13 Issue 2

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2023 > 13(2): 701-714

Next Article Previous Article

Changfeng Ma, Ting Wang. THE MODULUS-BASED MATRIX SPLITTING METHOD WITH INNER ITERATION FOR A CLASS OF NONLINEAR COMPLEMENTARITY PROBLEMS[J]. Journal of Applied Analysis & Computation, 2023, 13(2): 701-714. doi: 10.11948/20210515

Citation:

Changfeng Ma, Ting Wang. THE MODULUS-BASED MATRIX SPLITTING METHOD WITH INNER ITERATION FOR A CLASS OF NONLINEAR COMPLEMENTARITY PROBLEMS[J]. Journal of Applied Analysis & Computation, 2023, 13(2): 701-714. doi: 10.11948/20210515

THE MODULUS-BASED MATRIX SPLITTING METHOD WITH INNER ITERATION FOR A CLASS OF NONLINEAR COMPLEMENTARITY PROBLEMS

Changfeng Ma^1, ,,
Ting Wang²

1.
School of Big Data & Key Laboratory of Digital Technology and Intelligent Computing, Fuzhou University of International Studies and Trade, Fuzhou 350202, China
2.
School of Mathematics and Informatics, Fujian Normal University, Fuzhou 350117, China

Corresponding author: Email: mcf@fzfu.edu.cn(C. Ma)

Abstract

Abstract

In this paper, we propose a modulus-based matrix splitting iteration method with inner iteration for a class of nonlinear complementarity problems. Convergence conditions of the iteration method are analyzed carefully, which shows that the iteration sequence generated by this method converges to a solution of the NCP under certain conditions. Moreover, the convergence conditions of the proposed method are studied when the system matrix is symmetric positive definite or is an $ H_{+} $-matrix. Theoretical results are supported by the numerical experiments, which implies that the iteration method with inner iteration is more effective and feasible for solving certain nonlinear complementarity problems.
- Nonlinear complementarity problem /
- matrix splitting iteration method /
- modulus-based /
- convergence analysis
MSC: 90C33

FullText(HTML)

References

[1]	Z. Bai, On the monotone convergence of the projected iteration methods for linear complementarity problem, Numer. Math., J. Chin. Univer. (Eng. Ser.), 1996, 5, 228–233. Google Scholar
[2]	Z. Bai, On the convergence of the multisplitting methods for the linear complementarity problem, SIAM. J. Matrix Anal. Appl., 1999, 21, 67–78. doi: 10.1137/S0895479897324032 CrossRef Google Scholar
[3]	Z. Bai and D. Evans, Matrix multisplitting methods with applications to linear complementarity problems: parallel asychronous methods, Int. J. Comput. Math., 2002, 79, 205–232. doi: 10.1080/00207160211927 CrossRef Google Scholar
[4]	Z. Bai and L. Zhang, Modulus-based synchronous multisplitting iteration methods for linear complementarity problems, Numer. Linear Algebra Appl., 2013, 20, 425–439. doi: 10.1002/nla.1835 CrossRef Google Scholar
[5]	Z. Bai, Modulus-based matrix splitting iteration methods for linear complemenarity problems, Numer. Linear Algebra Appl., 2010, 17, 917–933. doi: 10.1002/nla.680 CrossRef Google Scholar
[6]	Z. Bai, The monotone convergence of a class of parallel nonlinear relaxation methods for nonlinear complementarity problems, Comput. Math. Appl., 1996, 31, 17–33. Google Scholar
[7]	Z. Bai, New comparison theorem for the nonlinear multisplitting relaxation methods for the large sparse nonlinear complementarity problems, Comput. Math. Appl., 1996, 32, 79–95. Google Scholar
[8]	L. Badea, X. Tai and J. Wang, Convergence rate analysis of a multiplicative Schwarz method for variational inequalities, SIAM J. Numer. Anal., 2003, 41, 1052–1073. doi: 10.1137/S0036142901393607 CrossRef Google Scholar
[9]	R. W. Cottle, J. Pang and R. E. Stone, The Linear Complementarity Problem, Academic, San Diego, 1992. Google Scholar
[10]	J. Dong and M. Jiang, A modified modulus method for symmetric positive-definite linear complementarity problems, Numer. Linear Algebra Appl., 2009, 16, 129–143. doi: 10.1002/nla.609 CrossRef Google Scholar
[11]	M. C. Ferris and J. Pang, Engineering and economic applications of complementarity problems, SIAM Review, 1997, 39, 669–713. doi: 10.1137/S0036144595285963 CrossRef Google Scholar
[12]	F. Facchinei and J. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer Science and Business Media, New York, 2003. Google Scholar
[13]	N. Huang and C. Ma, The modulus-based matrix splitting algorithms for a class of weakly nonlinear complementerity problems, Numer. Linear Algebra Appl., 2016, 23, 558–569. doi: 10.1002/nla.2039 CrossRef Google Scholar
[14]	J. Hong and C. Li, Modulus-based matrix splitting iteration methods for a class of implicit complementarity problems, Numer. Linear Algebra Appl., 2016, 23(4), 629–641. doi: 10.1002/nla.2044 CrossRef Google Scholar
[15]	Y. Jiang and J. Zeng, A Multiplicative Schwarz algorithm for the nonlinear complementarity problem with an $M$-function, Bull. Aust. Soc., 2010, 82, 353–366. doi: 10.1017/S0004972710000389 CrossRef $M$-function" target="_blank">Google Scholar
[16]	C. Kanzow, Inexact semismooth Newton methods for large-scale complementarity problems, Optim. Methods Softw., 2004, 19, 309–325. doi: 10.1080/10556780310001636369 CrossRef Google Scholar
[17]	B. M. Kwak and J. Y. Kwak, Binary NCP: a new approach for solving nonlinear complementarity problems, J. Mech. Sci. Tech., 2021, 35(3), 1161–1166. doi: 10.1007/s12206-021-0229-5 CrossRef Google Scholar
[18]	W. La Cruz, A spectral algorithm for large-scale systems of nonlinear monotone equations, Numer. Algor., 2017, 76(4), 1109–1130. doi: 10.1007/s11075-017-0299-8 CrossRef Google Scholar
[19]	F. Mezzadri and E. Galligani, Modulus-based matrix splitting methods for a class of horizontal nonlinear complementarity problems, Numer. Algor., 2020, 87(2), 667–687. Google Scholar
[20]	K. G. Murty, Linear Complementarity, Linear and Nonlinear Programming, Heldermann, Berlin, 1988. Google Scholar
[21]	G. H. Meyer, Free boundary problems with nonlinear source terms, Numer. Math., 1984, 43, 463–483. doi: 10.1007/BF01390185 CrossRef Google Scholar
[22]	C. Ma, L. Chen and D. Wang, A globally and superlinearly convergent smoothing Broyden-like method for solving nonlinear complementarity prolem, Appl. Math. Comput., 2008, 198, 592–604. Google Scholar
[23]	J. Pang, On the convergence of a basic iterative method for the implicit complementarity problems, J. Optim. Theory Appli., 1982, 37, 149–162. doi: 10.1007/BF00934765 CrossRef Google Scholar
[24]	Z. Xia and C. Li, Modulus-based matrix splitting iteration methods for a class of nonlinear complementerity problem, Appl. Math. Comput., 2015, 271, 34–42. Google Scholar
[25]	S. Xie and H. Xu, Two-step modulus-based matrix splitting iteration method for a class of nonlinear complementarity problems, Linear Algebra Appl., 2016, 494, 1–10. doi: 10.1016/j.laa.2016.01.002 CrossRef Google Scholar
[26]	S. Xie and H. Xu, An Efficient Class of Modulus-Based Matrix Splitting Methods for Nonlinear Complementarity Problems, Math. Prob. Engin., 2021, ID 9030547. Google Scholar
[27]	N. Zheng and J. Yin, Convergence of accelerated modulus-based matrix splitting iteration methods for linear complementarity problem with an $H_{+}-matrix$, J. Comput. Appl. Math., 2014, 260, 281–293. doi: 10.1016/j.cam.2013.09.079 CrossRef $H_{+}-matrix$" target="_blank">Google Scholar
[28]	L. Zhang, Two-step modulus-based matrix splitting iteration methods for linear complementarity problems, Numer Algor., 2011, 57, 83–99. doi: 10.1007/s11075-010-9416-7 CrossRef Google Scholar
[29]	L. Zhang and Z. Ren, Improved convergence theorems of modulus-based matrix splitting iteration methods for linear complementarity problems, Appl. Math. Lett., 2013, 26, 638–642. doi: 10.1016/j.aml.2013.01.001 CrossRef Google Scholar
[30]	L. Zhang, Two-stage multispliting iteration methods using modulus-based matrix splitting as inner iteration for linear complementarity problems, J. Optim. Theory Appl., 2014, 160, 189–203. doi: 10.1007/s10957-013-0362-0 CrossRef Google Scholar
[31]	Y. Zhang, Multilevel projection algorithm for solving obstacle problems, Comput. Math. Appl., 2001, 41, 1505–1513. doi: 10.1016/S0898-1221(01)00115-8 CrossRef Google Scholar
[32]	X. Zhang and Z. Peng, A modulus-based nonmonotone line search method for nonlinear complementarity problems, Appl. Math. Comput., 2020, 387, 125175. Google Scholar