Primal-dual algorithms for multi-agent structured optimization over message-passing architectures with bounded communication delays

We consider algorithms for solving structured convex optimization problems over a network of agents with communication delays. It is assumed that each agent performs its local updates by using possibly outdated information from its neighbors under the assumption that the delay with respect to each neighbor is bounded but otherwise arbitrary. The private objective of each agent is represented by the sum of two possibly nonsmooth functions, one of which is composed with a linear mapping. The global optimization problem is the aggregate of the local cost functions and a common Lipschitz-differentiable term. When the coupling between the agents is represented only through the common function the primal-dual algorithm proposed by V~u and Condat can be conveniently employed, while for more general structures a new algorithm is proposed. Moreover, a randomized variant is presented that allows the agents to wake up at random and independently from one another. The convergence of each of the proposed algorithms is established under different strong convexity assumptions.