Solving a class of asymmetric variational inequalities by a new alternating direction method

The augmented Lagrangian method (also referred to as an alternating directi on method) solves a class of variational inequalities (VI) via solving a se ries of sub-VI problems. The method is effective whenever the subproblems c an be solved efficiently. However, the subproblem to be solved in each iter ation of the augmented Lagrangian method itself is still a VI problem. It i s essentially as difficult as the original one, the only difference is that the dimension of the subproblems is lower, In this paper, we propose a new alternating direction method for solving a class of monotone variational i nequalities. In each iteration, the method solves a convex quadratic progra mming with simple constrains and a well-conditioned system of nonlinear equ ations. For such 'easier' subproblems, existing efficient numerical softwar es are applicable. The effectiveness of the proposed method is demonstrated with an illustrative example. (C) 2000 Elsevier Science Ltd. All rights re served.