A Series Of Projection And Contraction Alternating Direction Methods For Solving Separable Variational Inequality Problems

The alternating direction methods are attractive for solving separable variational inequality problems, which solve the original large-scale problems via solving a series of small-scale problems. However, the subproblems solved per iteration are still vari-ational inequality problems, which are structurally as difficult to solve as the original problems. In this paper, we propose a series of projection and contraction alternat-ing direction methods, which, per iteration, solve some projection problem and avoid solving the subproblems of variational inequality problems. As the series of methods, in this paper, we propose three new arithmetics. Under the same mild assumption as those in classical alternating direction methods, we prove global convergence of the new methods. We also present some preliminary numerical results, which show that our methods are sufficient.