An accelerated Newton method of high-order convergence for solving a class of weakly nonlinear complementarity problems
-
1833
Downloads
-
2899
Views
Authors
Ya-Jun Xie
- College of Mathematics and Informatics, Fujian Key Laboratory of Mathematical Analysis and Applications, Fujian Normal University, Fuzhou 350117, P. R. China.
- Department of Mathematics and Physics, Fujian Jiangxia University, Fuzhou 350108, P. R. China.
Na Huang
- College of Mathematics and Informatics, Fujian Key Laboratory of Mathematical Analysis and Applications, Fujian Normal University, Fuzhou 350117, P. R. China.
- Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, P. R. China.
Chang-Feng Ma
- College of Mathematics and Informatics, Fujian Key Laboratory of Mathematical Analysis and Applications, Fujian Normal University, Fuzhou 350117, P. R. China.
Abstract
In this paper, by extending the classical Newton method, we investigate an accelerated Newton iteration method (ANIM) with high-order convergence for solving a class of weakly nonlinear complementarity problems which arise from the discretization of free boundary problems. Theoretically, the performance of high-order convergence is analyzed in details. Some numerical experiments demonstrate the efficiency of the presented method.
Share and Cite
ISRP Style
Ya-Jun Xie, Na Huang, Chang-Feng Ma, An accelerated Newton method of high-order convergence for solving a class of weakly nonlinear complementarity problems, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 9, 4822--4833
AMA Style
Xie Ya-Jun, Huang Na, Ma Chang-Feng, An accelerated Newton method of high-order convergence for solving a class of weakly nonlinear complementarity problems. J. Nonlinear Sci. Appl. (2017); 10(9):4822--4833
Chicago/Turabian Style
Xie, Ya-Jun, Huang, Na, Ma, Chang-Feng. "An accelerated Newton method of high-order convergence for solving a class of weakly nonlinear complementarity problems." Journal of Nonlinear Sciences and Applications, 10, no. 9 (2017): 4822--4833
Keywords
- Weakly nonlinear complementarity problems
- high-order convergence
- the modulus-based nonlinear function
- convergence analysis
- numerical experience.
MSC
References
-
[1]
Z.-Z. Bai , On the convergence of the multisplitting methods for the linear complementarity problem, SIAM J. Matrix Anal. Appl., 21 (1999), 67–78.
-
[2]
Z.-Z. Bai, Experimental study of the asynchronous multisplitting relaxtation methods for linear complementarity problems, J. Comput. Math., 20 (2002), 561–574.
-
[3]
Z.-Z. Bai, Modulus-based matrix splitting iteration methods for linear complementarity problems, Numer. Linear Algebra Appl., 17 (2010), 917–933.
-
[4]
Z.-Z. Bai, D. J. Evans, Matrix multisplitting relaxtion methods for linear complementarity problem, Inter. J. Comput. Math., 63 (1997), 309–326.
-
[5]
Z.-Z. Bai, D. J. Evans, Matrix multisplitting methods with applications to linear complementarity problems: Parallel synchronous and chaotic methods, Calculateurs Parallelés Réseaux et Systémes Répartis, 13 (2001), 125–151.
-
[6]
Z.-Z. Bai, D. J. Evans, Parallel chaotic multisplitting iterative methods for the large spase linear complementarity problem, J. Comput. Math., 19 (2001), 281–292.
-
[7]
Z.-Z. Bai, D. J. Evans , Matrix multisplitting methods with applications to linear complementarity problems: Parallel asynchronous methods, Int. J. Comput. Math., 79 (2002), 205–232.
-
[8]
Z.-Z. Bai, Y.-G. Huang, A class of asynchronous iterations for the linear complementarity problem, J. Comput. Math., 21 (2003), 773–790.
-
[9]
Z.-Z. Bai, L.-L. Zhang, Modulus-based synchronous multisplitting iteration methods for linear complementarity problems, Numer. Linear Algebra Appl., 20 (2013), 425–439.
-
[10]
X. Chen, Z. Nashed, L. Qi , Smoothing methods and semismooth methods for nondifferentiable operator equations, SIAM J. Numer. Anal., 38 (2000), 1200–1216.
-
[11]
R. W. Cottle, J.-S. Pang, R. E. Stone, The Linear Complementarity Problem, Academic Press, Boston (1992)
-
[12]
C. Cryer , The solution of a quadratic programming using systematic overrelaxation, SIAM J. Control, 9 (1971), 385–392.
-
[13]
L. Cvetković, V. Kostić , A note on the convergence of the MSMAOR method for linear complementarity problems , Numer. Linear Algebra Appl., 21 (2014), 534–539.
-
[14]
B. C. Eaves, C. E. Lemke, Equivalence of LCP and PLS, Math. Oper. Res., 6 (1981), 475–484.
-
[15]
B. C. Eaves, H. Scarf , The solution of systems of piecewise linear equations, Math. Oper. Res., 1 (1976), 1–27.
-
[16]
Y. ElFoutayeni, M. Khaladi , Using vector divisions in solving the linear complementarity problem, J. Comput. Appl. Math., 236 (2012), 1919–1925.
-
[17]
A. Fischer, A special Newton-type optimization method, Optimization, 24 (1992), 269–284.
-
[18]
N. Huang, C.-F. Ma , The modulus-based matrix splitting algorithms for a class of weakly nonlinear complementarity problems, Numer. Linear Algebra Appl., 23 (2016), 558–569.
-
[19]
X.-D. Huang, Z.-G. Zeng, Y.-N. Ma , The theory and methods for nonlinear numerical analysis, Wuhan University Press, Wuhan (2004)
-
[20]
H.-Y. Jiang, L. Qi , A new nonsmooth equations approach to nonlinear complementarity problems, SIAM J. Control Optim., 35 (1997), 178–193.
-
[21]
C. E. Lemke , Bimatrix equilibrium points and mathematical programming, Management Sci., 11 (1965), 681–689.
-
[22]
D. Luca, F. Fancchinei, C. Kanzow, A semismooth equation approach to the solution of nonlinear complementarity problems, Math. Programming, 75 (1996), 407–439.
-
[23]
K. G. Murty , Linear Complementarity, Linear and Nonlinear Programming, Heldermann, Berlin (1988)
-
[24]
J.-S. Pang, Newton’s method for B-differentiable equations, Math. Oper. Res., 15 (1990), 311–341.
-
[25]
L. Qi, J. Sun, A nonsmooth version of Newtons method, Math. Programming, 58 (1993), 353–367.
-
[26]
H. Scarf, The approximation of fixed points of a continuous mapping, SIAM J. Appl. Math., 15 (1967), 1328–1343.
-
[27]
Z. Sun, J.-P. Zeng, A monotone semismooth Newton type method for a class of complementarity problems, J. Comput. Appl. Math., 235 (2011), 1261–1274.
-
[28]
P. Tseng, On linear convergence of iterative method for the variational inequality problem, J. Comput. Appl. Math., 60 (1995), 237–252.
-
[29]
Z. Yu, Y. Qin, A cosh-based smoothing Newton method for P0 nonlinear complementarity problem, Nonlinear Anal. Real World Appl., 12 (2011), 875–884.
-
[30]
L.-L. Zhang , Two-step modulus based matrix splitting iteration for linear complementarity problems, Numer. Algorithms, 57 (2011), 83–99.
-
[31]
N. Zheng, J.-F. Yin , Accelerated modulus-based matrix splitting iteration methods for linear complementarity problem, Numer. Algorithms, 64 (2013), 245–262.