On the Lie bracket approximation approach to distributed optimization: Extensions and limitations

TL;DR

This paper extends the Lie bracket approximation method for distributed convex optimization, enabling convergence to solutions in more general settings with broader classes of constraints and objective functions.

Contribution

It generalizes previous Lie bracket-based distributed optimization methods to handle more complex problems with general constraints and less restrictive assumptions.

Findings

Method converges to a neighborhood of the optimal solution

Applicable to a wider class of optimization problems

Works under mild communication topology assumptions

Abstract

We consider the problem of solving a smooth convex optimization problem with equality and inequality constraints in a distributed fashion. Assuming that we have a group of agents available capable of communicating over a communication network described by a time-invariant directed graph, we derive distributed continuous-time agent dynamics that ensure convergence to a neighborhood of the optimal solution of the optimization problem. Following the ideas introduced in our previous work, we combine saddle-point dynamics with Lie bracket approximation techniques. While the methodology was previously limited to linear constraints and objective functions given by a sum of strictly convex separable functions, we extend these result here and show that it applies to a very general class of optimization problems under mild assumptions on the communication topology.