  Alternating Direction Method of Multipliers (ADMM) is a popular convex
optimization algorithm, which can be employed for solving distributed consensus
optimization problems. In this setting agents locally estimate the optimal
solution of an optimization problem and exchange messages with their neighbors
over a connected network. The distributed algorithms are typically exposed to
different types of errors in practice, e.g., due to quantization or
communication noise or loss. We here focus on analyzing the convergence of
distributed ADMM for consensus optimization in presence of additive random node
error, in which case, the nodes communicate a noisy version of their latest
estimate of the solution to their neighbors in each iteration. We present
analytical upper and lower bounds on the mean squared steady state error of the
algorithm in case that the local objective functions are strongly convex and
have Lipschitz continuous gradients. In addition we show that, when the local
objective functions are convex and the additive node error is bounded, the
estimation error of the noisy ADMM for consensus optimization is also bounded.
Numerical results are provided which demonstrate the effectiveness of the
presented analyses and shed light on the role of the system and network
parameters on performance.
