  In this paper, a decentralized proximal method of multipliers (DPMM) is
proposed to solve constrained convex optimization problems over multi-agent
networks, where the local objective of each agent is a general closed convex
function, and the constraints are coupled equalities and inequalities. This
algorithm strategically integrates the dual decomposition method and the
proximal point algorithm. One advantage of DPMM is that subproblems can be
solved inexactly and in parallel by agents at each iteration, which relaxes the
restriction of requiring exact solutions to subproblems in many distributed
constrained optimization algorithms. We show that the first-order optimality
residual of the proposed algorithm decays to $0$ at a rate of $o(1/k)$ under
general convexity. Furthermore, if a structural assumption for the considered
optimization problem is satisfied, the sequence generated by DPMM converges
linearly to an optimal solution. In numerical simulations, we compare DPMM with
several existing algorithms using two examples to demonstrate its
effectiveness.
