On the Lie bracket approximation approach to distributed optimization: Extensions and limitations
For researchers in distributed optimization, this work incrementally extends prior methods to broader problem classes but does not introduce new paradigms or achieve SOTA results.
This paper extends Lie bracket approximation-based distributed optimization to general convex problems with equality and inequality constraints, proving convergence to a neighborhood of the optimal solution under mild assumptions on directed communication graphs.
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.