A LQP BASED INTERIOR PREDICTION-CORRECTION METHOD FOR NONLINEAR COMPLEMENTARITY PROBLEMS 1)

To solve nonlinear complementarity problems (NCP), at each iteration, the classical proximal point algorithm solves a well-conditioned sub-NCP while the LogarithmicQuadratic Proximal (LQP) method solves a system of nonlinear equations (LQP system). This paper presents a practical LQP method-based prediction-correction method for NCP. The predictor is obtained via solving the LQP system approximately under significantly relaxed restriction, and the new iterate (the corrector) is computed directly by an explicit formula derived from the original LQP method. The implementations are very easy to be carried out. Global convergence of the method is proved under the same mild assumptions as the original LQP method. Finally, numerical results for traffic equilibrium problems are provided to verify that the method is effective for some practical problems.

[1]  Alfredo N. Iusem,et al.  A Generalized Proximal Point Algorithm for the Variational Inequality Problem in a Hilbert Space , 1998, SIAM J. Optim..

[2]  Bingsheng He,et al.  Inexact implicit methods for monotone general variational inequalities , 1999, Math. Program..

[3]  A. Nagurney.  Network Economics: A Variational Inequality Approach , 1992 .

[4]  Andreas Fischer,et al.  Solution of monotone complementarity problems with locally Lipschitzian functions , 1997, Math. Program..

[5]  Jonathan Eckstein,et al.  Approximate iterations in Bregman-function-based proximal algorithms , 1998, Math. Program..

[6]  A. Nagurney,et al.  Projected Dynamical Systems and Variational Inequalities with Applications , 1995 .

[7]  R. Rockafellar.  Monotone Operators and the Proximal Point Algorithm , 1976 .

[8]  L. Liao,et al.  Improvements of Some Projection Methods for Monotone Nonlinear Variational Inequalities , 2002 .

[9]  Marc Teboulle,et al.  Convergence of Proximal-Like Algorithms , 1997, SIAM J. Optim..

[10]  Alfredo N. Iusem,et al.  An interior point method with Bregman functions for the variational inequality problem with paramonotone operators , 1998, Math. Program..

[11]  Hai Yang,et al.  TRAFFIC RESTRAINT, ROAD PRICING AND NETWORK EQUILIBRIUM , 1997 .

[12]  Marc Teboulle,et al.  Lagrangian Duality and Related Multiplier Methods for Variational Inequality Problems , 1999, SIAM J. Optim..

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

[14]  Marc Teboulle,et al.  A Logarithmic-Quadratic Proximal Method for Variational Inequalities , 1999, Comput. Optim. Appl..

[15]  Mounir Haddou,et al.  An interior-proximal method for convex linearly constrained problems and its extension to variational inequalities , 1995, Math. Program..

[16]  Osman Güer.  On the convergence of the proximal point algorithm for convex minimization , 1991 .