New decomposition methods for solving variational inequality problems

For solving large-scale constrained separable variational inequality problems, the decomposition methods are attractive, since they solve the original problems via solving a series of small-scale problems, which may be much easier to solve than the original problems. In this paper, we propose some new decomposition methods, which are based on the Lagrange and the augmented Lagrange mappings of the problems, respectively. For the global convergence, the first method needs the partial cocoercivity of the underlying mapping, while the second one just requires monotonicity, a condition which is much weaker than partial cocoercivity. The cost for this weaker condition is to perform two additional projection steps on the dual variables and the primal-dual variables. We then extend the method to a more practical one, which just solves the subproblem approximately. We also report some computational results of the inexact method to show its promise.