Approximate Versions of the Alternating Direction Method of Multipliers∗

We present three new approximate versions of alternating direction method of multipliers (ADMM), all of which require only knowledge of subgradients of the subproblem objectives, rather than bounds on the distance to the exact subproblem solution. One version, which applies only to certain common special cases, is based on combining the operator-splitting analysis of the ADMM with a relative-error proximal point algorithm of Solodov and Svaiter. A byproduct of this analysis is a new, relative-error version of the Douglas-Rachford splitting algorithm for monotone operators. The other two approximate versions of the ADMM are more general and based on the Lagrangian splitting analysis of the ADMM: one uses a summable absolute error criterion, and the other uses a relative error criterion and an auxiliary iterate sequence. We experimentally compare our new algorithms to an essentially exact form of the ADMM and to an inexact form that can be easily derived from prior theory (but again applies only to certain common special cases). These experiments show that our methods can significantly reduce total computational effort when iterative methods are used to solve ADMM subproblems.