An inexact logarithmic-quadratic proximal augmented Lagrangian method for a class of constrained variational inequalities

The augmented Lagrangian method is attractive in constraint optimizations. When it is applied to a class of constrained variational inequalities, the sub-problem in each iteration is a nonlinear complementarity problem (NCP). By introducing a logarithmic-quadratic proximal term, the sub-NCP becomes a system of nonlinear equations, which we call the LQP system. Solving a system of nonlinear equations is easier than the related NCP, because the solution of the NCP has combinatorial properties. In this paper, we present an inexact logarithmic-quadratic proximal augmented Lagrangian method for a class of constrained variational inequalities, in which the LQP system is solved approximately under a rather relaxed inexactness criterion. The generated sequence is Fejér monotone and the global convergence is proved. Finally, some numerical test results for traffic equilibrium problems are presented to demonstrate the efficiency of the method.

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

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

[3]  Stephen J. Wright,et al.  Numerical Optimization , 2018, Fundamental Statistical Inference.

[4]  G. M. Korpelevich.  The extragradient method for finding saddle points and other problems , 1976 .

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

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

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

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

[9]  Bingsheng He,et al.  A LQP BASED INTERIOR PREDICTION-CORRECTION METHOD FOR NONLINEAR COMPLEMENTARITY PROBLEMS 1) , 2006 .

[10]  John N. Tsitsiklis,et al.  Parallel and distributed computation , 1989 .

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

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

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

[14]  E. Khobotov.  Modification of the extra-gradient method for solving variational inequalities and certain optimization problems , 1989 .

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

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