This paper investigates a distributed algorithm for the multi-agent constrained optimization problem, which is to minimize a global objective function formed by a sum of local convex (possibly nonsmooth) functions under both coupled inequality and affine equality constraints. By introducing auxiliary variables, we decouple the constraints and transform the multi-agent optimization problem into a {variational inequality problem} with a set-valued monotone mapping. We propose a distributed dual averaging algorithm to find the weak solutions of the variational inequality problem with an $O(1/\sqrt{k})$ convergence rate, {where $k$ is the number of iterations.} Moreover, we show that weak solutions are also strong solutions that match the optimal primal-dual solutions to the considered optimization problem. {A numerical example is} given for illustration.
distributed optimization, variational inequality, coupled constraints, dual averaging, multi-agent networks
90C33, 68W15