A Levenberg–Marquardt Method for Nonlinear Complementarity Problems Based on Nonmonotone Trust Region and Line Search Techniques

Using the FB function, we propose a new Levenberg–Marquardt algorithm for nonlinear complementarity problem. To obtain the global convergence, the algorithm uses the nonmonotone trust region and line search techniques under a convenient boundedness assumption. Furthermore, we get local superlinear/quadratic convergence of the algorithm under a nonsingularity condition. Some numerical examples are given to illustrate the performance and efficiency of the presented algorithm.

[1]  Kenneth Levenberg.  A METHOD FOR THE SOLUTION OF CERTAIN NON – LINEAR PROBLEMS IN LEAST SQUARES , 1944 .

[2]  D. Marquardt.  An Algorithm for Least-Squares Estimation of Nonlinear Parameters , 1963 .

[3]  F. Clarke.  Generalized gradients and applications , 1975 .

[4]  M. Powell.  CONVERGENCE PROPERTIES OF A CLASS OF MINIMIZATION ALGORITHMS , 1975 .

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

[6]  John E. Dennis,et al.  Numerical methods for unconstrained optimization and nonlinear equations , 1983, Prentice Hall series in computational mathematics.

[7]  Jong-Shi Pang,et al.  Nonsmooth Equations: Motivation and Algorithms , 1993, SIAM J. Optim..

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

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

[10]  J. M. Martínez,et al.  Solving Nonsmooth Equations by Means of Quasi-Newton Methods with Globalization , 1995 .

[11]  S. Dirkse,et al.  Mcplib: a collection of nonlinear mixed complementarity problems , 1995 .

[12]  Masao Fukushima,et al.  Modified Newton methods for solving a semismooth reformulation of monotone complementarity problems , 1996, Math. Program..

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

[14]  Christian Kanzow,et al.  Global Convergence Properties of Some Iterative Methods for Linear Complementarity Problems , 1996, SIAM J. Optim..

[15]  Patrick T. Harker,et al.  Smooth Approximations to Nonlinear Complementarity Problems , 1997, SIAM J. Optim..

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

[17]  Xiaojun Chen,et al.  Global and superlinear convergence of the smoothing Newton method and its application to general box constrained variational inequalities , 1998, Math. Comput..

[18]  Jorge Nocedal,et al.  Combining Trust Region and Line Search Techniques , 1998 .

[19]  D. Gao.  Analytic solutions and triality theory for nonconvex and nonsmooth variational problems with applications , 2000 .

[20]  Defeng Sun,et al.  A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities , 2000, Math. Program..

[21]  M. Fukushima,et al.  On the Rate of Convergence of the Levenberg-Marquardt Method , 2001 .

[22]  Masao Fukushima,et al.  Convergence Properties of the Inexact Levenberg-Marquardt Method under Local Error Bound Conditions , 2002, Optim. Methods Softw..

[23]  Jin-yanFan.  A MODIFIED LEVENBERG-MARQUARDT ALGORITHM FOR SINGULAR SYSTEM OF NONLINEAR EQUATIONS , 2003 .

[24]  William W. Hager,et al.  A Nonmonotone Line Search Technique and Its Application to Unconstrained Optimization , 2004, SIAM J. Optim..

[25]  Liqun Qi,et al.  Smoothing Trust Region Methods for Nonlinear Complementarity Problems with P0-Functions , 2005, Ann. Oper. Res..

[26]  Shuning Wang,et al.  Global optimization of separable objective functions on convex polyhedra via piecewise-linear approximation , 2006 .

[27]  Ju-Liang Zhang,et al.  A smoothing Levenberg-Marquardt method for NCP , 2006, Appl. Math. Comput..

[28]  Jia Tang,et al.  A globally convergent Levenberg-Marquardt method for solving nonlinear complementarity problem , 2007, Appl. Math. Comput..

[29]  Zhijian Huang,et al.  Numerical experience with newton-like methods for nonlinear algebraic systems , 1997, Computing.

[30]  Changfeng Ma,et al.  A new smoothing and regularization Newton method for P0-NCP , 2010, J. Glob. Optim..

[31]  Yan Gao,et al.  Convergence analysis of nonmonotone Levenberg-Marquardt algorithms for complementarity problem , 2010, Appl. Math. Comput..

[32]  Li,et al.  A SMOOTHING TRUST REGION METHOD FOR NCPS BASED ON THE SMOOTHING GENERALIZED FISCHER-BURMEISTER FUNCTION * , 2010 .

[33]  Changfeng Ma,et al.  A smoothing Levenberg-Marquardt algorithm for solving a class of stochastic linear complementarity problem , 2011, Appl. Math. Comput..

[34]  Changfeng Ma,et al.  A smoothing Newton method for solving a class of stochastic linear complementarity problems , 2011 .

[35]  D. Pu,et al.  Smoothing Levenberg–Marquardt method for general nonlinear complementarity problems under local error bound , 2011 .

[36]  Changfeng Ma,et al.  Superlinear/quadratic smoothing Broyden-like method for the generalized nonlinear complementarity problem , 2011 .

[37]  Changfeng Ma,et al.  A new smoothing Broyden-like method for solving nonlinear complementarity problem with a P0-function , 2011, J. Glob. Optim..