  In this work, we consider the 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 Distributed Alternating Direction Method of
Multipliers using Finite-Time Exact Ratio Consensus (D-ADMM-FTERC), to solve
the multi-node convex optimization problem, in which every node performs
iterative computations and exchanges information with its neighbors. At every
iteration of D-ADMM-FTERC, each node solves a local convex optimization problem
for the one of the primal variables and utilizes a finite-time exact consensus
protocol to obtain the optimal value of the other variable, since the cost
function for the second primal variable is not decomposable. Since D-ADMM-FTERC
requires to know the upper bound on the number of nodes in the network, we
furthermore propose a new algorithm, called Fully D-ADMM Finite-Time
Distributed Termination (FD-ADMM-FTDT) algorithm, which does not need any
global information. If the individual cost functions are convex and
not-necessarily differentiable, the proposed algorithms converge at a rate of
O(1/k), where k is the iteration counter. Additionally, if the global objective
function is strongly convex and smooth, the proposed algorithms have an
"approximate" R-linear convergence rate. The efficacy of FD-ADMM-FTDT is
demonstrated via a distributed L1 regularized logistic regression optimization
example. Additionally, comparisons with other state-of-the-art algorithms are
provided on large-scale networks showing the superior precision and
time-efficient performance of FD-ADMM-FTDT.
