  This paper studies efficient distributed optimization methods for multi-agent
networks. Specifically, we consider a convex optimization problem with a
globally coupled linear equality constraint and local polyhedra constraints,
and develop distributed optimization methods based on the alternating direction
method of multipliers (ADMM). The considered problem has many applications in
machine learning and smart grid control problems. Due to the presence of the
polyhedra constraints, agents in the existing methods have to deal with
polyhedra constrained subproblems at each iteration. One of the key issues is
that projection onto a polyhedra constraint is not trivial, which prohibits
from closed-form solutions or the use of simple algorithms for solving these
subproblems. In this paper, by judiciously integrating the proximal
minimization method with ADMM, we propose a new distributed optimization method
where the polyhedra constraints are handled softly as penalty terms in the
subproblems. This makes the subproblems efficiently solvable and consequently
reduces the overall computation time. Furthermore, we propose a randomized
counterpart that is robust against randomly ON/OFF agents and imperfect
communication links. We analytically show that both the proposed methods have a
worst-case $\mathcal{O}(1/k)$ convergence rate, where $k$ is the iteration
number. Numerical results show that the proposed methods offer considerably
lower computation time than the existing distributed ADMM method.
