On the Step Size of Symmetric Alternating Directions Method of Multipliers

The alternating direction method of multipliers (ADMM) is an application of the Douglas-Rachford splitting method; and the symmetric version of ADMM which updates the Lagrange multiplier twice at each iteration is an application of the Peaceman-Rachford splitting method. Sometimes the symmetric ADMM works empirically; but theoretically its convergence is not guaranteed. It was recently found that the convergence of symmetric ADMM can be sufficiently ensured if both the step sizes for updating the Lagrange multiplier are shrank conservatively. In this paper, we focus on the convex programming context and specify a step size domain that can ensure the convergence of symmetric ADMM. In particular, it is shown for the first time that, we can choose Glowinski’s larger step size for one of the Lagrange-multiplier-updating steps at each iteration; so shrinking both the step sizes is not necessary for the symmetric ADMM. Some known results in the ADMM literature turn out to be special cases of our discussion.