  This article reports an algorithm for multi-agent distributed optimization
problems with a common decision variable, local linear equality and inequality
constraints and set constraints with convergence rate guarantees.
\textcolor{black}{The algorithm accrues all the benefits of the Alternating
Direction Method of Multipliers (ADMM) approach}. It also overcomes the
limitations of existing methods on convex optimization problems with linear
inequality, equality and set constraints by allowing directed communication
topologies. Moreover, the algorithm can be synthesized distributively. The
developed algorithm has: (i) a $O(1/k)$ rate of convergence, where $k$ is the
iteration counter, when individual functions are convex but not-necessarily
differentiable, and (ii) a geometric rate of convergence to any arbitrary small
neighborhood of the optimal solution, when the objective functions are smooth
and restricted strongly convex at the optimal solution. The efficacy of the
algorithm is evaluated by a comparison with state-of-the-art constrained
optimization algorithms in solving a constrained distributed
$\ell_1$-regularized logistic regression problem, and unconstrained
optimization algorithms in solving a $\ell_1$-regularized Huber loss
minimization problem. Additionally, a comparison of the algorithm's performance
with other algorithms in the literature that utilize multiple communication
steps is provided.
