Periodic and event-triggered communication for distributed continuous-time convex optimization

We propose a distributed continuous-time algorithm to solve a network optimization problem where the global cost function is a strictly convex function composed of the sum of the local cost functions of the agents. We establish that our algorithm, when implemented over strongly connected and weight-balanced directed graph topologies, converges exponentially fast when the local cost functions are strongly convex and their gradients are globally Lipschitz. We also characterize the privacy preservation properties of our algorithm and extend the convergence guarantees to the case of time-varying, strongly connected, weight-balanced digraphs. When the network topology is a connected undirected graph, we show that exponential convergence is still preserved if the gradients of the strongly convex local cost functions are locally Lipschitz, while it is asymptotic if the local cost functions are convex. We also study discrete-time communication implementations. Specifically, we provide an upper bound on the stepsize of a synchronous periodic communication scheme that guarantees convergence over connected undirected graph topologies and, building on this result, design a centralized event-triggered implementation that is free of Zeno behavior. Simulations illustrate our results.