  In this paper, we develop a novel distributed algorithm for addressing convex
optimization with both nonlinear inequality and linear equality constraints,
where the objective function can be a general nonsmooth convex function and all
the constraints can be fully coupled. Specifically, we first separate the
constraints into three groups, and design two primal-dual methods and utilize a
virtual-queue-based method to handle each group of the constraints
independently. Then, we integrate these three methods in a strategic way,
leading to an integrated primal-dual proximal (IPLUX) algorithm, and enable the
distributed implementation of IPLUX. We show that IPLUX achieves an $O(1/k)$
rate of convergence in terms of optimality and feasibility, which is stronger
than the convergence results of the state-of-the-art distributed algorithms for
convex optimization with coupling nonlinear constraints. Finally, IPLUX
exhibits competitive practical performance in the simulations.
