On the O(1/k) Convergence Rate of He’s Alternating Directions Method for a Kind of Structured Variational Inequality Problem

The alternating directions method for a kind of structured variational inequality problem (He, 2001) is an attractive method for structured monotone variational inequality problems. In each iteration, the subproblems are convex quadratic minimization problem with simple constraints and a well-conditioned system of nonlinear equations that can be efficiently solved using classical methods. Researchers have recently described the convergence rate of projection and contraction methods for variational inequality problems and the original ADM and its linearized variant. Motivated and inspired by research into the convergence rate of these methods, we provide a simple proof to show the O(1/k) convergence rate of alternating directions methods for structured monotone variational inequality problems (He, 2001).