A Continuous-Time Gradient-Tracking Algorithm for Directed Networks

In this letter, we consider the problem of unconstrained convex optimization over directed networks and design a continuous-time (CT) gradient-tracking dynamics to address it. First, we establish that the optimum of the distributed optimization problem (DOP) is contained in the equilibrium of the designed dynamics under appropriate initialization. Subsequently, we construct a novel Lyapunov function to establish exponential convergence to the equilibrium. Specifically, for the Lyapunov function, we rely on the Lyapunov-like equations associated with the asymmetric graph Laplacians of the directed networks. As a result of the convergence analysis, we obtain sufficiency conditions on the gains involved in the dynamics. Additionally, we also present an adaptive variant of the designed gradient-tracking dynamics which converges to the aforementioned equilibrium asymptotically.