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

The augmented Lagrangian method (also referred to as an alternating direction method) solves a class of variational inequalities (VI) via solving a series of sub-VI problems. The method is effective whenever the subproblems can be solved efficiently. However, the subproblem to be solved in each iteration of the augmented Lagrangian method itself is still a VI problem. It is 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 inequalities. In each iteration, the method solves a convex quadratic programming with simple constrains and a well-conditioned system of nonlinear equations. For such ‘easier’ subproblems, existing efficient numerical softwares are applicable. The effectiveness of the proposed method is demonstrated with an illustrative example.