The modulus-based nonsmooth Newton's method for solving linear complementarity problems

As applying the nonsmooth Newton's method to the equivalent reformulation of the linear complementarity problem, a modulus-based nonsmooth Newton's method is established and its locally quadratical convergence conditions are presented. In the implementation, local one step convergence is discussed by properly choosing the initial vector and the generalized Jacobian, and a mixed algorithm is given for finding an initial vector. Numerical experiments show that the proposed methods are efficient and accelerate the convergence performance of the modulus-based matrix splitting iteration methods.

[1]  Some approximate methods of solving operator equations , 1972 .

[2]  Apostolos Hadjidimos,et al.  On Iterative Solution for Linear Complementarity Problem with an H+-Matrix , 2012, SIAM J. Matrix Anal. Appl..

[3]  Xiaojun Chen,et al.  Newton-like methods for solving underdetermined nonlinear equations with nondifferentiable terms , 1994 .

[4]  Y. Ye,et al.  On Homotopy-Smoothing Methods for Box-Constrained Variational Inequalities , 1999 .

[5]  Jerzy Stefan Respondek,et al.  Numerical recipes for the high efficient inverse of the confluent Vandermonde matrices , 2011, Appl. Math. Comput..

[6]  Richard Bellman,et al.  Introduction to matrix analysis (2nd ed.) , 1997 .

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

[8]  Apostolos Hadjidimos,et al.  Nonstationary Extrapolated Modulus Algorithms for the solution of the Linear Complementarity Problem , 2009 .

[9]  Elisa Schae.  Review of a course on the web graph by Anthony Bonato (American Mathematical Society, Providence, Rhode Island, USA) , 2009, SIGA.

[10]  Tetsuro Yamamoto,et al.  A note on a posteriori error bound of zabrejko and nguen for zincenko's iteration , 1987 .

[11]  Xiaojun Chen.  SMOOTHING METHODS FOR COMPLEMENTARITY PROBLEMS AND THEIR APPLICATIONS : A SURVEY , 2000 .

[12]  Xiaojun Chen,et al.  On the convergence of some quasi-Newton methods for nonlinear equations with nondifferentiable operators , 2005, Computing.

[13]  Lidia Aceto,et al.  On the A-stable methods in the GBDF class , 2002 .

[14]  Jerzy S. Respondek.  Recursive numerical recipes for the high efficient inversion of the confluent Vandermonde matrices , 2013, Appl. Math. Comput..

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

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

[17]  David W. Lewis,et al.  Matrix theory , 1991 .

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

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

[20]  Richard Bellman,et al.  Introduction to Matrix Analysis , 1972 .

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

[22]  Xiaojun Chen,et al.  Ball-convergence theorems and error estimates for certain iterative methods for nonlinear equations , 1990 .

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

[24]  Y. Ye,et al.  On Homotopy-Smoothing Methods for Variational Inequalities , 1999 .

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

[26]  D. Trigiante,et al.  The stability problem for linear multistep methods: Old and new results , 2007 .

[27]  R. D. Murphy,et al.  Iterative solution of nonlinear equations , 1994 .

[28]  James M. Ortega,et al.  Iterative solution of nonlinear equations in several variables , 2014, Computer science and applied mathematics.

[29]  Byong-Hun Ahn,et al.  Iterative methods for linear complementarity problems with upperbounds on primary variables , 1983, Mathematical programming.

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

[31]  F. Facchinei,et al.  Finite-Dimensional Variational Inequalities and Complementarity Problems , 2003 .

[32]  Xiaojun Chen,et al.  On the convergence of Broyden-like methods for nonlinear equations with nondifferentiable terms , 1990 .

[33]  Xiaojun Chen,et al.  A parameterized Newton method and a quasi-Newton method for nonsmooth equations , 1994, Comput. Optim. Appl..

[34]  F. Clarke.  Optimization And Nonsmooth Analysis , 1983 .