Distributed Nonconvex Optimization with Exponential Convergence Rate via Hybrid Systems Methods

TL;DR

This paper introduces a hybrid systems approach for distributed nonconvex optimization, achieving exponential convergence rates by combining continuous-time computation with discrete communication, applicable to multi-agent systems.

Contribution

It develops a novel hybrid systems framework that guarantees global exponential convergence for nonconvex optimization under the PL condition, integrating continuous and discrete dynamics.

Findings

Proves global exponential stability of the proposed system.

Demonstrates effectiveness through simulations in three applications.

Shows impact of initial conditions on convergence rates.

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 three different applications and illustrate the convergence rates and the impact of initial conditions upon convergence.