Linearly Convergent Asynchronous Distributed ADMM via Markov Sampling

We consider the consensual distributed optimization problem and propose an asynchronous version of the Alternating Direction Method of Multipliers (ADMM) algorithm to solve it. The `asynchronous' part here refers to the fact that only one node/processor is updated (i.e. performs a minimization step) at each iteration of the algorithm. The selection of the node to be updated is decided by simulating a Markov chain. The proposed algorithm is shown to have a linear convergence property in expectation for the class of functions which are strongly convex and continuously differentiable.