A Modified Distributed Gradient Dynamics for Multiagent Optimization on Directed Networks

This article considers the distributed convex optimization problem over directed multiagent networks. We introduce a continuous-time coordination algorithm to solve unconstrained optimization problems with additive structure. The proposed algorithm can be interpreted as a modified version of the distributed subgradient method, enhanced with an augmented scalar state variable. Each agent is assumed to know its out-degree. Unlike existing methods that rely on a perturbed version of the push-sum algorithm, the proposed algorithm does not require any specific initialization. As a result, it is capable of handling strongly connected networks with sporadically varying sizes. We show that the proposed network flow is guaranteed to converge to the global minimizer of a sum of convex functions, provided that the local objective functions are strongly convex and have Lipschitz-continuous gradients. In addition, by considering a class of admissible time-varying gains/step-sizes, our analysis substantiates an explicit sublinear rate of convergence for the proposed algorithm.