Abstract: Alternating direction methods are suitable ones for solving large-scale problems. This paper presents a new alternating direction method for a class of variational inequalities. At each iteration, the proposed subproblem consists of a strongly monotonic linear variational inequality and a well-conditioned system of nonlinear equations, which is easily to be solved. The convergence theorem of the proposed method is proved based on the exact solution of the subproblem. Furthermore, the authors develop the proposed alternating direction method as an inexact method, which only needs to solve the subproblem inexactly. Under some inexact conditions, the convergence of inexact alternating direction method is proved too.

Key words: Variational inequality, Alternating direction method, Inexact method,
Convergence