Modified modulus-based matrix splitting algorithms for a class of weakly nondifferentiable nonlinear complementarity problems

By reformulating a class of weakly nonlinear complementarity problems as implicit fixed-point equations based on splitting of the system matrix, a modified modulus-based matrix splitting algorithm is presented. The convergence analysis of proposed algorithm is established for the case that the splitting of the system matrix is an H-splitting. Numerical experiments on two model problems are given to illustrate the theoretical results and examine the numerical effectiveness.

[1]  Changfeng Ma,et al.  On convergence of a smoothing Broyden-like method for P0-NCP☆ , 2008 .

[2]  M. Fiedler,et al.  On matrices with non-positive off-diagonal elements and positive principal minors , 1962 .

[3]  Na Huang,et al.  A new extragradient-like method for solving variational inequality problems , 2012 .

[4]  Alfredo N. Iusem,et al.  Convergence of direct methods for paramonotone variational inequalities , 2010, Comput. Optim. Appl..

[5]  C. Kanzow Some equation-based methods for the nonlinear complementarity problem , 1994 .

[6]  Patrick T. Harker,et al.  Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications , 1990, Math. Program..

[7]  Changfeng Ma,et al.  A globally and superlinearly convergent smoothing Broyden-like method for solving nonlinear complementarity problem , 2008, Appl. Math. Comput..

[8]  Katta G. Murty,et al.  Linear complementarity, linear and nonlinear programming , 1988 .

[9]  Zhong-Zhi Bai,et al.  The convergence of parallel iteration algorithms for linear complementarity problems , 1996 .

[10]  Richard S. Varga,et al.  Matrix Iterative Analysis , 2000, The Mathematical Gazette.

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

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

[13]  Muhammad Aslam Noor Iterative methods for a class of complementarity problems , 1988 .

[14]  Daniel P. Robinson,et al.  Subspace Accelerated Matrix Splitting Algorithms for Asymmetric and Symmetric Linear Complementarity Problems , 2013, SIAM J. Optim..

[15]  Changfeng Ma,et al.  A regularization smoothing Newton method for solving nonlinear complementarity problem , 2009 .

[16]  C. M. Elliott,et al.  Weak and variational methods for moving boundary problems , 1982 .

[17]  Changfeng Ma,et al.  The convergence of a one-step smoothing Newton method for P0-NCP based on a new smoothing NCP-function , 2008 .

[18]  Gunter H. Meyer,et al.  Free boundary problems with nonlinear source terms , 1984 .

[19]  J. Y. Bello Cruz,et al.  A Strongly Convergent Direct Method for Monotone Variational Inequalities in Hilbert Spaces , 2009 .

[20]  P. Tseng On linear convergence of iterative methods for the variational inequality problem , 1995 .

[21]  Jong-Shi Pang,et al.  NE/SQP: A robust algorithm for the nonlinear complementarity problem , 1993, Math. Program..

[22]  Changfeng Ma,et al.  Parallel multisplitting iteration methods based on M-splitting for the PageRank problem , 2015, Appl. Math. Comput..

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

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

[25]  B. Ahn Solution of nonsymmetric linear complementarity problems by iterative methods , 1981 .

[26]  Changfeng Ma,et al.  The convergence of a smoothing damped Gauss-Newton method for nonlinear complementarity problem , 2009 .

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

[28]  Francisco Facchinei,et al.  A semismooth equation approach to the solution of nonlinear complementarity problems , 1996, Math. Program..

[29]  Mei-Qun Jiang,et al.  A modified modulus method for symmetric positive‐definite linear complementarity problems , 2009, Numer. Linear Algebra Appl..

[30]  A. Iusem,et al.  A variant of korpelevich’s method for variational inequalities with a new search strategy , 1997 .

[31]  O. Mangasarian Solution of symmetric linear complementarity problems by iterative methods , 1977 .

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

[33]  K.-H. Hoffmann,et al.  Parallel solution of variational inequality problems with nonlinear source terms , 1996 .

[34]  Zhe Sun,et al.  A monotone semismooth Newton type method for a class of complementarity problems , 2011, J. Comput. Appl. Math..

[35]  C. Cryer The Solution of a Quadratic Programming Problem Using Systematic Overrelaxation , 1971 .

[36]  Yair Censor,et al.  The Subgradient Extragradient Method for Solving Variational Inequalities in Hilbert Space , 2011, J. Optim. Theory Appl..

[37]  Wei-wei Xu Modified modulus-based matrix splitting iteration methods for linear complementarity problems , 2015, Numer. Linear Algebra Appl..

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

[39]  Changfeng Ma,et al.  The numerical study of a regularized smoothing Newton method for solving P0-NCP based on the generalized smoothing Fischer-Burmeister function , 2012, Appl. Math. Comput..

[40]  Houyuan Jiang,et al.  A New Nonsmooth Equations Approach to Nonlinear Complementarity Problems , 1997 .

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