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In this paper, we establish the convergence properties for a majorizedalternating direction method of multipliers (ADMM) for linearly constrainedconvex optimization problems whose objectives contain coupled functions. Ourconvergence analysis relies on the generalized Mean-Value Theorem which playsan important role to properly control the cross terms due to the presence ofcoupled objective functions. Our results in particular show that directlyapplying 2-block ADMM with a large step length to the linearly constrainedconvex optimization problem with a quadratically coupled objective function isconvergent under mild conditions. We also provide several iteration complexityresults for the algorithm.

On the convergence properties of a majorized ADMM for linearly constrained convex optimization problems with coupled objective functions