A non-modulus linear method for solving the linear complementarity problem

Abstract In this paper, a non-modulus linear method for solving the linear complementarity problem is established by using the sign patterns of the solution of the equivalent modulus equation. In the proposed method the efficient numerical algorithms for solving the linear equations can be applied to the large sparse problems. Numerical examples show that the new method is valid.

[1]  Y. Saad,et al.  GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems , 1986 .

[2]  David J. Evans,et al.  Matrix multisplitting relaxation methods for linear complementarity problems , 1997, Int. J. Comput. Math..

[3]  Stephen J. Wright.  A path-following interior-point algorithm for linear and quadratic problems , 1996, Ann. Oper. Res..

[4]  Wen Li,et al.  The modulus-based nonsmooth Newton's method for solving linear complementarity problems , 2015, J. Comput. Appl. Math..

[5]  Li-Li Zhang,et al.  Modulus‐based synchronous multisplitting iteration methods for linear complementarity problems , 2013, Numer. Linear Algebra Appl..

[6]  Ahn Byong-hun.  Iterative methods for linear complementarity problems with upperbounds on primary variables , 1983 .

[7]  Zhong-Zhi Bai,et al.  A modified damped Newton method for linear complementarity problems , 2006, Numerical Algorithms.

[8]  Li-Li Zhang,et al.  Modulus-based synchronous two-stage multisplitting iteration methods for linear complementarity problems , 2012, Numerical Algorithms.

[9]  Li-Li Zhang,et al.  Two-step modulus-based matrix splitting iteration method for linear complementarity problems , 2011, Numerical Algorithms.

[10]  Van Bokhoven.  Piecewise-linear modelling and analysis , 1981 .

[11]  M. Hestenes,et al.  Methods of conjugate gradients for solving linear systems , 1952 .

[12]  Hua Zheng,et al.  A preconditioned modulus-based iteration method for solving linear complementarity problems of H-matrices , 2016 .

[13]  Yinyu Ye,et al.  On quadratic and $$O\left( {\sqrt {nL} } \right)$$ convergence of a predictor—corrector algorithm for LCP , 1993, Math. Program..

[14]  Zhong-Zhi Bai,et al.  On the monotone convergence of matrix multisplitting relaxation methods for the linear complementarity problem , 1998 .

[15]  Ning Zheng,et al.  Accelerated modulus-based matrix splitting iteration methods for linear complementarity problem , 2012, Numerical Algorithms.

[16]  Liqun Qi,et al.  A nonsmooth version of Newton's method , 1993, Math. Program..

[17]  David J. Evans,et al.  Matrix Multisplitting Methods with Applications to Linear Complementarity Problems∶ Parallel Asynchronous Methods , 2002, Int. J. Comput. Math..

[18]  Richard W. Cottle,et al.  Linear Complementarity Problem. , 1992 .

[19]  Stephen J. Wright,et al.  A superquadratic infeasible-interior-point method for linear complementarity problems , 1994, Math. Program..

[20]  Wen Li,et al.  A general modulus-based matrix splitting method for linear complementarity problems of HH-matrices , 2013, Appl. Math. Lett..

[21]  Zhong-Zhi Bai,et al.  On the Convergence of the Multisplitting Methods for the Linear Complementarity Problem , 1999, SIAM J. Matrix Anal. Appl..

[22]  Li-Li Zhang.  Two-Stage Multisplitting Iteration Methods Using Modulus-Based Matrix Splitting as Inner Iteration for Linear Complementarity Problems , 2014, J. Optim. Theory Appl..

[23]  Li-Li Zhang,et al.  Improved convergence theorems of modulus-based matrix splitting iteration methods for linear complementarity problems , 2013, Appl. Math. Lett..

[24]  Naonori Kakimura.  Sign-solvable linear complementarity problems , 2008 .

[25]  Robert J. Plemmons,et al.  Nonnegative Matrices in the Mathematical Sciences , 1979, Classics in Applied Mathematics.

[26]  Li-Li Zhang,et al.  Two-step modulus-based synchronous multisplitting iteration methods for linear complementarity problems , 2015 .

[27]  Hua Zheng,et al.  A general accelerated modulus-based matrix splitting iteration method for solving linear complementarity problems , 2016 .

[28]  Zhong-Zhi Bai,et al.  Modulus‐based matrix splitting iteration methods for linear complementarity problems , 2010, Numer. Linear Algebra Appl..

[29]  Kevin A. McShane.  Superlinearly Convergent O(√(n) L)-Iteration Interior-Point Algorithms for Linear Programming and the Monotone Linear Complementarity Problem , 1994, SIAM J. Optim..

[30]  Stephen J. Wright,et al.  Local convergence of interior-point algorithms for degenerate monotone LCP , 1994, Comput. Optim. Appl..

[31]  Michael C. Ferris,et al.  Engineering and Economic Applications of Complementarity Problems , 1997, SIAM Rev..