A Lie bracket approximation approach to distributed optimization over directed graphs

TL;DR

This paper introduces a novel distributed continuous-time optimization method over directed graphs using Lie bracket approximation, enabling cooperative solving of constrained problems with minimal network assumptions.

Contribution

It develops a new Lie bracket-based approach for designing distributed optimization algorithms over directed graphs, relaxing previous topological and constraint assumptions.

Findings

Effective in directed graph settings

Handles shared linear constraints

Simplifies in special cases

Abstract

We consider a group of computation units trying to cooperatively solve a distributed optimization problem with shared linear equality and inequality constraints. Assuming that the computation units are communicating over a network whose topology is described by a time-invariant directed graph, by combining saddle-point dynamics with Lie bracket approximation techniques we derive a methodology that allows to design distributed continuous-time optimization algorithms that solve this problem under minimal assumptions on the graph topology as well as on the structure of the constraints. We discuss several extensions as well as special cases in which the proposed procedure becomes particularly simple.