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Two-Stage Multisplitting Iteration Methods Using Modulus-Based Matrix Splitting as Inner Iteration for Linear Complementarity Problems

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Abstract

The matrix multisplitting iteration method is an effective tool for solving large sparse linear complementarity problems. However, at each iteration step we have to solve a sequence of linear complementarity sub-problems exactly. In this paper, we present a two-stage multisplitting iteration method, in which the modulus-based matrix splitting iteration and its relaxed variants are employed as inner iterations to solve the linear complementarity sub-problems approximately. The convergence theorems of these two-stage multisplitting iteration methods are established. Numerical experiments show that the two-stage multisplitting relaxation methods are superior to the matrix multisplitting iteration methods in computing time, and can achieve a satisfactory parallel efficiency.

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Notes

  1. MAOR, MSOR, MGS and MJ are the abbreviations of the modulus-based AOR, SOR, Gauss–Seidel and Jacobi methods for solving the linear complementarity problems; see [8].

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Acknowledgements

The author thanks the referees and the editor for their constructive suggestions and helpful comments that led to significant improvement of the original manuscript of this paper.

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  1. Li-Li Zhang

    Present address: Department of Mathematics and Information Science, Henan University of Economics and Law, Zhengzhou, 450002, P.R. China

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  1. State Key Laboratory of Scientific/Engineering Computing, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, P.R. China

    Li-Li Zhang

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Correspondence to Li-Li Zhang.

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Communicated by Michael Patriksson.

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Zhang, LL. Two-Stage Multisplitting Iteration Methods Using Modulus-Based Matrix Splitting as Inner Iteration for Linear Complementarity Problems. J Optim Theory Appl 160, 189–203 (2014). https://doi.org/10.1007/s10957-013-0362-0

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