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

H. Saberi Najafi; S. A. Edalatpanah

RAIRO - Operations Research - Recherche Opérationnelle (2013)

  • Volume: 47, Issue: 1, page 59-71
  • ISSN: 0399-0559

Abstract

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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.

How to cite

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Saberi Najafi, H., and Edalatpanah, S. A.. "Iterative methods with analytical preconditioning technique to linear complementarity problems: application to obstacle problems." RAIRO - Operations Research - Recherche Opérationnelle 47.1 (2013): 59-71. <http://eudml.org/doc/275034>.

@article{SaberiNajafi2013,
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.},
author = {Saberi Najafi, H., Edalatpanah, S. A.},
journal = {RAIRO - Operations Research - Recherche Opérationnelle},
keywords = {linear complementarity problems; preconditioning; iterative methods; H-matrix; obstacle problems; -matrix},
language = {eng},
number = {1},
pages = {59-71},
publisher = {EDP-Sciences},
title = {Iterative methods with analytical preconditioning technique to linear complementarity problems: application to obstacle problems},
url = {http://eudml.org/doc/275034},
volume = {47},
year = {2013},
}

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
PB - EDP-Sciences
VL - 47
IS - 1
SP - 59
EP - 71
AB - 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.
LA - eng
KW - linear complementarity problems; preconditioning; iterative methods; H-matrix; obstacle problems; -matrix
UR - http://eudml.org/doc/275034
ER -

References

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  1. [1] C.E. Lemke, Bimatrix equilibrium points and mathematical programming. Manag. Sci.11 (1965) 681–689. Zbl0139.13103MR189823
  2. [2] C.W. Cryer, The solution of a quadratic programming problem using systematic overrelaxation. SIAM J. Control.9 (1971) 385–392. Zbl0216.54603MR298922
  3. [3] K.G. Murty, Linear Complementarity, Linear and Nonlinear Programming. Heldermann Verlag, Berlin (1988). Zbl0634.90037MR949214
  4. [4] M.S. Bazaraa, H.D. Sherali and C.M. Shetty, Nonlinear programming, Theory and algorithms, 3rd ed., Hoboken, NJ: Wiley-Interscience (2006). Zbl0774.90075MR2218478
  5. [5] R.W. Cottle, J.S. Pang and R.E. Stone, The Linear Complementarity Problem. Academic Press, London (1992). Zbl0757.90078
  6. [6] R.W. Cottle and G.B. Dantzig, Complementarity pivot theory of mathematical programming. Linear Algebra Appl.1 (1968) 103–125. Zbl0155.28403
  7. [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. Zbl0482.90084
  8. [8] O.L. Mangasarian, Solution of symmetric linear complementarity problems by iterative methods. J. Optim. Theory Appl.22 (1977) 465–485. Zbl0341.65049
  9. [9] O.L. Mangasarian, Sparsity preserving SOR algorithms for separable quadratic and linear programming. Comput. Oper. Res.11 (1984) 105–112. Zbl0606.90102
  10. [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. Zbl0595.90090
  11. [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. Zbl0777.90070MR1094794
  12. [12] R. De Leone and O.L. Mangasarian, Asynchronous parallel successive Overrelaxation for the Symmetric linear complementarity problem. Math. Progr.42 (1988) 347–361. Zbl0665.90090MR976125
  13. [13] Z.Z. Bai and D.J. Evans, Matrix multisplitting relaxation methods for linear complementarity Problems. Int. J. Comput. Math.63 (1997) 309–326. Zbl0876.90086MR1455239
  14. [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. Zbl0942.65059MR1709726
  15. [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. Zbl1011.65033MR1892468
  16. [16] D. Yuan and Y.Z. Song, Modified AOR methods for linear complementarity problem. Appl. Math. Comput.140 (2003) 53–67. Zbl1030.65065MR1944115
  17. [17] Y. Li and P. Dai, Generalized AOR methods for linear complementarity problem. Appl. Math. Comput.188 (2007) 7–18. Zbl1121.65068MR2327089
  18. [18] K.R. James, Convergence of matrix iterations subject to diagonal dominance. SIAM. J. Numer. Anal.12 (1973) 478–484. Zbl0255.65019MR334476
  19. [19] Y.Z. Song, On the convergence of the generalized AOR method. Linear Algebra Appl.256 (1997) 199–218. Zbl0872.65028MR1438234
  20. [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. Zbl1262.90176MR3022313
  21. [21] R.S. Varga, Matrix Iterative Analysis, 2nd ed., Springer, Berlin (2000). Zbl0998.65505MR1753713
  22. [22] A. Frommer and D.B. Szyld, H–splitting and two–stage iterative methods. Numer. Math.63 (1992) 345–356. Zbl0764.65018MR1186346
  23. [23] A. Berman and R.J. Plemmons, Nonnegative Matrices in the Mathematical Sciences. 3rd ed., SIAM, Philadelphia (1994). Zbl0815.15016MR1298430
  24. [24] J.P. Milaszewicz, Improving Jacobi and Gauss-Seidel iterations. Linear Algebra Appl.93 (1987) 161–170. Zbl0628.65022MR898552
  25. [25] A. Gunawardena, S. Jain and L. Snyder, Modified iterative methods for consistent linear systems. Linear Algebra Appl. 154-156 (1991) 123–143. Zbl0731.65016MR1113142
  26. [26] M. Usui, H. Niki and T. Kohno, Adaptive Gauss Seidel method for linear systems. Int. J. Comput. Math.51 (1994) 119–125. Zbl0828.65028
  27. [27] H. Saberi Najafi and S.A. Edalatpanah, A New Family of (I+S)-type Preconditioner with Some Applications. Submitted manuscript. Zbl1331.65058
  28. [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. [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. Zbl1181.65049MR2557284
  30. [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. Zbl0992.65022MR1840641
  31. [31] L. Wang and Y. Song, preconditioned AOR iterative method for M-matrices. J. Comput. Appl. Math. 226 (2009) 114–124. Zbl1175.65043MR2501886
  32. [32] A. Hadjidimos, Accelerated Overrelaxation method. Math. Comput.32 (1978) 149–157. Zbl0382.65015MR483340
  33. [33] D.J. Evans and M.M. Martins, On the convergence of the extrapolated AOR method. Int. J. Comput. Math.43 (1992) 161–171. Zbl0754.65032
  34. [34] B. Truyen and J. Cornelis, Adiabatic Layering: A New Concept of Hierarchical Muliti-scale Optimization. Neural Networks8 (1995) 1373–1378. 
  35. [35] P.G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland Publishing Company, Amsterdam (1978). Zbl0511.65078MR520174

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