Iterative methods with analytical preconditioning technique to linear complementarity problems: application to obstacle problems
RAIRO - Operations Research - Recherche Opérationnelle, Tome 47 (2013) no. 1, pp. 59-71.

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.

DOI : 10.1051/ro/2013027
Classification : 90C33, 65F10
Mots clés : linear complementarity problems, preconditioning, iterative methods, H-matrix, obstacle problems
@article{RO_2013__47_1_59_0,
     author = {Saberi Najafi, H. and Edalatpanah, S. A.},
     title = {Iterative methods with analytical preconditioning technique to linear complementarity problems: application to obstacle problems},
     journal = {RAIRO - Operations Research - Recherche Op\'erationnelle},
     pages = {59--71},
     publisher = {EDP-Sciences},
     volume = {47},
     number = {1},
     year = {2013},
     doi = {10.1051/ro/2013027},
     mrnumber = {3143742},
     zbl = {1276.90075},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ro/2013027/}
}
TY  - JOUR
AU  - Saberi Najafi, H.
AU  - Edalatpanah, S. A.
TI  - Iterative methods with analytical preconditioning technique to linear complementarity problems: application to obstacle problems
JO  - RAIRO - Operations Research - Recherche Opérationnelle
PY  - 2013
SP  - 59
EP  - 71
VL  - 47
IS  - 1
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/ro/2013027/
DO  - 10.1051/ro/2013027
LA  - en
ID  - RO_2013__47_1_59_0
ER  - 
%0 Journal Article
%A Saberi Najafi, H.
%A Edalatpanah, S. A.
%T Iterative methods with analytical preconditioning technique to linear complementarity problems: application to obstacle problems
%J RAIRO - Operations Research - Recherche Opérationnelle
%D 2013
%P 59-71
%V 47
%N 1
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/ro/2013027/
%R 10.1051/ro/2013027
%G en
%F RO_2013__47_1_59_0
Saberi Najafi, H.; Edalatpanah, S. A. Iterative methods with analytical preconditioning technique to linear complementarity problems: application to obstacle problems. RAIRO - Operations Research - Recherche Opérationnelle, Tome 47 (2013) no. 1, pp. 59-71. doi : 10.1051/ro/2013027. http://www.numdam.org/articles/10.1051/ro/2013027/

[1] C.E. Lemke, Bimatrix equilibrium points and mathematical programming. Manag. Sci. 11 (1965) 681-689. | MR | Zbl

[2] C.W. Cryer, The solution of a quadratic programming problem using systematic overrelaxation. SIAM J. Control. 9 (1971) 385-392. | MR | Zbl

[3] K.G. Murty, Linear Complementarity, Linear and Nonlinear Programming. Heldermann Verlag, Berlin (1988). | MR | Zbl

[4] M.S. Bazaraa, H.D. Sherali and C.M. Shetty, Nonlinear programming, Theory and algorithms, 3rd ed., Hoboken, NJ: Wiley-Interscience (2006). | MR | Zbl

[5] R.W. Cottle, J.S. Pang and R.E. Stone, The Linear Complementarity Problem. Academic Press, London (1992). | Zbl

[6] R.W. Cottle and G.B. Dantzig, Complementarity pivot theory of mathematical programming. Linear Algebra Appl. 1 (1968) 103-125. | Zbl

[7] J.S. Pang, On the convergence of a basic iterative method for the implicit complementarity problem. J. Optim. Theory Appl. 37 (1982) 149-162. | Zbl

[8] O.L. Mangasarian, Solution of symmetric linear complementarity problems by iterative methods. J. Optim. Theory Appl. 22 (1977) 465-485. | Zbl

[9] O.L. Mangasarian, Sparsity preserving SOR algorithms for separable quadratic and linear programming. Comput. Oper. Res. 11 (1984) 105-112. | Zbl

[10] O.L. Mangasarian and R. De Leone, Parallel successive Overrelaxation methods for symmetric Linear complementarity problems and linear programs. J. Optim. Theory Appl. 54 (1987) 437-446. | Zbl

[11] O.L. Mangasarian, Convergence of iterates of an inexact matrix splitting algorithm for the symmetric monotone linear complementarity problem. SIAM J. Optim. 1 (1991) 114-122. | MR | Zbl

[12] R. De Leone and O.L. Mangasarian, Asynchronous parallel successive Overrelaxation for the Symmetric linear complementarity problem. Math. Progr. 42 (1988) 347-361. | MR | Zbl

[13] Z.Z. Bai and D.J. Evans, Matrix multisplitting relaxation methods for linear complementarity Problems. Int. J. Comput. Math. 63 (1997) 309-326. | MR | Zbl

[14] Z.Z. Bai, On the convergence of the multisplitting methods for the linear complementarity Problem. SIAM J.Matrix Anal. Appl. 21 (1999) 67-78. | MR | Zbl

[15] Z.Z. Bai and D.J. Evans, Matrix multisplitting methods with applications to linear complementarity problems: parallel asynchronous methods. Int. J. Comput. Math. 79 (2002) 205-232. | MR | Zbl

[16] D. Yuan and Y.Z. Song, Modified AOR methods for linear complementarity problem. Appl. Math. Comput. 140 (2003) 53-67. | MR | Zbl

[17] Y. Li and P. Dai, Generalized AOR methods for linear complementarity problem. Appl. Math. Comput. 188 (2007) 7-18. | MR | Zbl

[18] K.R. James, Convergence of matrix iterations subject to diagonal dominance. SIAM. J. Numer. Anal. 12 (1973) 478-484. | MR | Zbl

[19] Y.Z. Song, On the convergence of the generalized AOR method. Linear Algebra Appl. 256 (1997) 199-218. | MR | Zbl

[20] 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. | MR | Zbl

[21] R.S. Varga, Matrix Iterative Analysis, 2nd ed., Springer, Berlin (2000). | MR | Zbl

[22] A. Frommer and D.B. Szyld, H-splitting and two-stage iterative methods. Numer. Math. 63 (1992) 345-356. | MR | Zbl

[23] A. Berman and R.J. Plemmons, Nonnegative Matrices in the Mathematical Sciences. 3rd ed., SIAM, Philadelphia (1994). | MR | Zbl

[24] J.P. Milaszewicz, Improving Jacobi and Gauss-Seidel iterations. Linear Algebra Appl. 93 (1987) 161-170. | MR | Zbl

[25] A. Gunawardena, S. Jain and L. Snyder, Modified iterative methods for consistent linear systems. Linear Algebra Appl. 154-156 (1991) 123-143. | MR | Zbl

[26] M. Usui, H. Niki and T. Kohno, Adaptive Gauss Seidel method for linear systems. Int. J. Comput. Math. 51 (1994) 119-125. | Zbl

[27] H. Saberi Najafi and S.A. Edalatpanah, A New Family of (I+S)-type Preconditioner with Some Applications. Submitted manuscript.

[28] H. Saberi Najafi and S.A. Edalatpanah, Some Improvements In PMAOR Method For Solving Linear Systems. J. Info. Comp. Sci. 6 (2011) 15-22.

[29] B. Zheng and S.X. Miao, Two new modified Gauss-Seidel methods for linear system with M-matrices. J. Comput. Appl. Math. 233 (2009) 922-930. | MR | Zbl

[30] D.J. Evans, M.M. Martins and M.E. Trigo, The AOR iterative method for new preconditioned linear systems. J. Comput. Appl. Math. 132 (2001) 461-466. | MR | Zbl

[31] L. Wang and Y. Song, preconditioned AOR iterative method for M-matrices. J. Comput. Appl. Math. 226 (2009) 114-124. | MR | Zbl

[32] A. Hadjidimos, Accelerated Overrelaxation method. Math. Comput. 32 (1978) 149-157. | MR | Zbl

[33] D.J. Evans and M.M. Martins, On the convergence of the extrapolated AOR method. Int. J. Comput. Math. 43 (1992) 161-171. | Zbl

[34] B. Truyen and J. Cornelis, Adiabatic Layering: A New Concept of Hierarchical Muliti-scale Optimization. Neural Networks 8 (1995) 1373-1378.

[35] P.G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland Publishing Company, Amsterdam (1978). | MR | Zbl

Cité par Sources :