Distributed Nonconvex Optimization with Exponential Convergence Rate via Hybrid Systems Methods

TL;DR

It is proved that this hybrid systems framework for distributed multi-agent optimization in which agents execute computations in continuous time and communicate in discrete time is globally exponentially stable to a minimizer of a possibly nonconvex, smooth objective function that satisfies the Polyak-Lojasiewicz (PL) condition.

Abstract

We present a hybrid systems framework for distributed multi-agent optimization in which agents execute computations in continuous time and communicate in discrete time. The optimization algorithm is analogous to a continuous-time form of parallelized coordinate descent. Agents implement an update-and-hold strategy in which gradients are computed at communication times and held constant during flows between communications. The completeness of solutions under these hybrid dynamics is established. Then, we prove that this system is globally exponentially stable to a minimizer of a possibly nonconvex, smooth objective function that satisfies the Polyak-Lojasiewicz (PL) condition. Simulation results are presented for four different applications and illustrate the convergence rates and the impact of initial conditions upon convergence.