Convergence of a smoothing-type algorithm for the monotone affine variational inequality problem

Smoothing-type algorithms have been applied to solve various optimization problems. In the analysis on the global convergence, most existing smoothing-type algorithms need to assume that the solution set of the problem concerned is nonempty and bounded or some stronger conditions. In this paper, we investigate a smoothing-type algorithm for solving the monotone affine variational inequality problem (AVIP). Specially, we reformulate the AVIP as a system of parameterized smooth equations, and instead of solving the original AVIP, we use a Newton-type method to solve the smooth equations. We show that under mild assumptions, the iteration sequence generated by the algorithm is bounded; and the algorithm may find a maximally complementary solution to the AVIP. In our analysis on the convergence, we do not need to assume that the solution set of the AVIP is bounded.

[1]  Defeng Sun,et al.  Sub-quadratic convergence of a smoothing Newton algorithm for the P0– and monotone LCP , 2004, Math. Program..

[2]  James V. Burke,et al.  The Global Linear Convergence of a Noninterior Path-Following Algorithm for Linear Complementarity Problems , 1998, Math. Oper. Res..

[3]  Yahya FATHI.  Computational complexity of LCPs associated with positive definite symmetric matrices , 1979, Math. Program..

[4]  Christian Kanzow,et al.  Predictor-Corrector Smoothing Methods for Linear Programs with a More Flexible Update of the Smoothing Parameter , 2002, Comput. Optim. Appl..

[5]  Song Xu,et al.  A non–interior predictor–corrector path following algorithm for the monotone linear complementarity problem , 2000, Math. Program..

[6]  Zhenghai Huang,et al.  Predictor-Corrector Smoothing Newton Method, Based on a New Smoothing Function, for Solving the Nonlinear Complementarity Problem with a P0 Function , 2003 .

[7]  Jie Sun,et al.  A smoothing Newton algorithm for the LCP with a sufficient matrix that terminates finitely at a maximally complementary solution , 2006, Optim. Methods Softw..

[8]  Christian Kanzow,et al.  Improved smoothing-type methods for the solution of linear programs , 2002, Numerische Mathematik.

[9]  Zheng-Hai Huang,et al.  A Non-Interior Continuation Algorithm for the P0 or P* LCP with Strong Global and Local Convergence Properties , 2005 .

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

[11]  Paul Tseng,et al.  Non-Interior continuation methods for solving semidefinite complementarity problems , 2003, Math. Program..

[12]  Olvi L. Mangasarian,et al.  A class of smoothing functions for nonlinear and mixed complementarity problems , 1996, Comput. Optim. Appl..

[13]  Masao Fukushima,et al.  A Combined Smoothing and Regularization Method for Monotone Second-Order Cone Complementarity Problems , 2005, SIAM J. Optim..

[14]  Defeng Sun,et al.  A Smoothing Newton-Type Algorithm of Stronger Convergence for the Quadratically Constrained Convex Quadratic Programming , 2006, Comput. Optim. Appl..

[15]  Zheng-Hai Huang,et al.  Locating a maximally complementary solution of the monotone NCP by using non-interior-point smoothing algorithms , 2005, Math. Methods Oper. Res..

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

[17]  Zheng-Hai Huang,et al.  Convergence properties of a non-interior-point smoothing algorithm for the P*NCP , 2007 .

[18]  Shu-Cherng Fang,et al.  On the Finite Termination of an Entropy Function Based Non-Interior Continuation Method for Vertical Linear Complementarity Problems , 2005, J. Glob. Optim..

[19]  Houduo Qi,et al.  A Regularized Smoothing Newton Method for Box Constrained Variational Inequality Problems with P0-Functions , 1999, SIAM J. Optim..

[20]  Christian Kanzow,et al.  Semidefinite Programs: New Search Directions, Smoothing-Type Methods, and Numerical Results , 2002, SIAM J. Optim..

[21]  Xiaojun Chen,et al.  A Global and Local Superlinear Continuation-Smoothing Method for P0 and R0 NCP or Monotone NCP , 1999, SIAM J. Optim..

[22]  Duan Li,et al.  A Globally and Locally Superlinearly Convergent Non--Interior-Point Algorithm for P[sub 0] LCPs , 2002, SIAM J. Optim..

[23]  N. Xiu,et al.  Superlinear Noninterior One-Step Continuation Method for Monotone LCP in the Absence of Strict Complementarity , 2001 .