  In this work, we consider the asynchronous distributed optimization problem
in which each node has its own convex cost function and can communicate
directly only with its neighbors, as determined by a directed communication
topology (directed graph or digraph). First, we reformulate the optimization
problem so that Alternating Direction Method of Multipliers (ADMM) can be
utilized. Then, we propose an algorithm, herein called Asynchronous Approximate
Distributed Alternating Direction Method of Multipliers (AsyAD-ADMM), using
finite-time asynchronous approximate ratio consensus, to solve the multi-node
convex optimization problem, in which every node performs iterative
computations and exchanges information with its neighbors asynchronously. More
specifically, at every iteration of AsyAD-ADMM, each node solves a local convex
optimization problem for one of the primal variables and utilizes a finite-time
asynchronous approximate consensus protocol to obtain the value of the other
variable which is close to the optimal value, since the cost function for the
second primal variable is not decomposable. If the individual cost functions
are convex but not necessarily differentiable, the proposed algorithm converges
at a rate of $\mathcal{O}(1/k)$, where $k$ is the iteration counter. The
efficacy of AsyAD-ADMM is exemplified via a proof-of-concept distributed
least-square optimization problem with different performance-influencing
factors investigated.
