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Iterative methods with analytical preconditioning technique to linear complementarity problems: application to obstacle problems

Published online by Cambridge University Press:  07 March 2013

H. Saberi Najafi  and
S.A. Edalatpanah
H. Saberi Najafi
Affiliation:
Department of Mathematics, Faculty of Sciences, Islamic Azad University, Lahijan Branch, Iran. hnajafi@guilan.ac.ir
S.A. Edalatpanah
Affiliation:
Department of Mathematics, Faculty of Sciences, Islamic Azad University, Lahijan Branch, Iran. hnajafi@guilan.ac.ir Young Researchers Club, Lahijan Branch, Islamic Azad University, Lahijan, Iran; saedalatpanah@gmail.com
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Abstract

For solving linear complementarity problems LCP more attention has recently been paid on a class of iterative methods called the matrix-splitting. But up to now, no paper has discussed the effect of preconditioning technique for matrix-splitting methods in LCP. So, this paper is planning to fill in this gap and we use a class of preconditioners with generalized Accelerated Overrelaxation (GAOR) methods and analyze the convergence of these methods for LCP. Furthermore, Comparison between our methods and other non-preconditioned methods for the studied problem shows a remarkable agreement and reveals that our models are superior in point of view of convergence rate and computing efficiency. Besides, by choosing the appropriate parameters of these methods, we derive same results as the other iterative methods such as AOR, JOR, SOR etc. Finally the method is tested by some numerical experiments.

Keywords

Linear complementarity problemspreconditioningiterative methodsH-matrixobstacle problems

Information

Type
Research Article
Information
RAIRO - Operations Research , Volume 47 , Issue 1 , January 2013 , pp. 59 - 71
Copyright
© EDP Sciences, ROADEF, SMAI, 2013

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References

Lemke, C.E., Bimatrix equilibrium points and mathematical programming. Manag. Sci. 11 (1965) 681–689. Google Scholar
Cryer, C.W., The solution of a quadratic programming problem using systematic overrelaxation. SIAM J. Control. 9 (1971) 385–392. Google Scholar
K.G. Murty, Linear Complementarity, Linear and Nonlinear Programming. Heldermann Verlag, Berlin (1988).
M.S. Bazaraa, H.D. Sherali and C.M. Shetty, Nonlinear programming, Theory and algorithms, 3rd ed., Hoboken, NJ: Wiley-Interscience (2006).
R.W. Cottle, J.S. Pang and R.E. Stone, The Linear Complementarity Problem. Academic Press, London (1992).
Cottle, R.W. and Dantzig, G.B., Complementarity pivot theory of mathematical programming. Linear Algebra Appl. 1 (1968) 103–125. Google Scholar
Pang, J.S., On the convergence of a basic iterative method for the implicit complementarity problem. J. Optim. Theory Appl. 37 (1982) 149–162. Google Scholar
Mangasarian, O.L., Solution of symmetric linear complementarity problems by iterative methods. J. Optim. Theory Appl. 22 (1977) 465–485. Google Scholar
Mangasarian, O.L., Sparsity preserving SOR algorithms for separable quadratic and linear programming. Comput. Oper. Res. 11 (1984) 105–112. Google Scholar
Mangasarian, O.L. and De Leone, R., Parallel successive Overrelaxation methods for symmetric Linear complementarity problems and linear programs. J. Optim. Theory Appl. 54 (1987) 437–446. Google Scholar
Mangasarian, O.L., Convergence of iterates of an inexact matrix splitting algorithm for the symmetric monotone linear complementarity problem. SIAM J. Optim. 1 (1991) 114–122. Google Scholar
De Leone, R. and Mangasarian, O.L., Asynchronous parallel successive Overrelaxation for the Symmetric linear complementarity problem. Math. Progr. 42 (1988) 347–361. Google Scholar
Bai, Z.Z. and Evans, D.J., Matrix multisplitting relaxation methods for linear complementarity Problems. Int. J. Comput. Math. 63 (1997) 309–326. Google Scholar
Bai, Z.Z., On the convergence of the multisplitting methods for the linear complementarity Problem. SIAM J.Matrix Anal. Appl. 21 (1999) 67–78. Google Scholar
Bai, Z.Z. and Evans, D.J., Matrix multisplitting methods with applications to linear complementarity problems: parallel asynchronous methods. Int. J. Comput. Math. 79 (2002) 205–232. Google Scholar
Yuan, D. and Song, Y.Z., Modified AOR methods for linear complementarity problem. Appl. Math. Comput. 140 (2003) 53–67. Google Scholar
Li, Y. and Dai, P., Generalized AOR methods for linear complementarity problem. Appl. Math. Comput. 188 (2007) 7–18. Google Scholar
James, K.R., Convergence of matrix iterations subject to diagonal dominance. SIAM. J. Numer. Anal. 12 (1973) 478–484. Google Scholar
Song, Y.Z., On the convergence of the generalized AOR method. Linear Algebra Appl. 256 (1997) 199–218. Google Scholar
H. Saberi Najafi and S.A. Edalatpanah, On the Convergence Regions of Generalized Accelerated Overrelaxation Method for Linear Complementarity Problems. J. Optim. Theory Appl. (2012) DOI:10.1007/s10957-012-0135-1.
R.S. Varga, Matrix Iterative Analysis, 2nd ed., Springer, Berlin (2000).
Frommer, A. and Szyld, D.B., H–splitting and two–stage iterative methods. Numer. Math. 63 (1992) 345–356. Google Scholar
A. Berman and R.J. Plemmons, Nonnegative Matrices in the Mathematical Sciences. 3rd ed., SIAM, Philadelphia (1994).
Milaszewicz, J.P., Improving Jacobi and Gauss-Seidel iterations. Linear Algebra Appl. 93 (1987) 161–170. Google Scholar
Gunawardena, A., Jain, S. and Snyder, L., Modified iterative methods for consistent linear systems. Linear Algebra Appl. 154-156 (1991) 123–143. Google Scholar
Usui, M., Niki, H. and Kohno, T., Adaptive Gauss Seidel method for linear systems. Int. J. Comput. Math. 51 (1994) 119–125. Google Scholar
H. Saberi Najafi and S.A. Edalatpanah, A New Family of (I+S)-type Preconditioner with Some Applications. Submitted manuscript.
Saberi Najafi, H. and Edalatpanah, S.A., Some Improvements In PMAOR Method For Solving Linear Systems. J. Info. Comp. Sci. 6 (2011) 15–22. Google Scholar
Zheng, B. and Miao, S.X., Two new modified Gauss-Seidel methods for linear system with M-matrices. J. Comput. Appl. Math. 233 (2009) 922–930. Google Scholar
Evans, D.J., Martins, M.M. and Trigo, M.E., The AOR iterative method for new preconditioned linear systems. J. Comput. Appl. Math. 132 (2001) 461–466. Google Scholar
Wang, L. and Song, Y., preconditioned AOR iterative method for M-matrices. J. Comput. Appl. Math. 226 (2009) 114–124. Google Scholar
Hadjidimos, A., Accelerated Overrelaxation method. Math. Comput. 32 (1978) 149–157. Google Scholar
Evans, D.J. and Martins, M.M., On the convergence of the extrapolated AOR method. Int. J. Comput. Math. 43 (1992) 161–171. Google Scholar
Truyen, B. and Cornelis, J., Adiabatic Layering: A New Concept of Hierarchical Muliti-scale Optimization. Neural Networks 8 (1995) 1373–1378. Google Scholar
P.G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland Publishing Company, Amsterdam (1978).