We design and analyze a fully distributed algorithm for convex constrained optimization in networks without any consistent naming infrastructure. The algorithm produces an approximately feasible and near-optimal solution in time polynomial in the network size, the inverse of the permitted error, and a measure of curvature variation in the dual optimization problem. It blends, in a novel way, gossip-based information spreading, iterative gradient ascent, and the barrier method from the design of interior-point algorithms.
