Reaching an Optimal Consensus: Dynamical Systems that Compute Intersections of Convex Sets

TL;DR

This paper presents a decentralized continuous-time algorithm for multi-agent systems to compute the intersection of convex sets, achieving consensus and optimal agreement under certain connectivity conditions.

Contribution

It introduces a simple distributed control rule that ensures convergence to the intersection of convex sets in multi-agent systems with time-varying topologies.

Findings

Achieves global optimal consensus in multi-agent systems.

Provides connectivity conditions for directed and bidirectional communications.

Establishes properties of distance functions related to the solution set.

Abstract

In this paper, multi-agent systems minimizing a sum of objective functions, where each component is only known to a particular node, is considered for continuous-time dynamics with time-varying interconnection topologies. Assuming that each node can observe a convex solution set of its optimization component, and the intersection of all such sets is nonempty, the considered optimization problem is converted to an intersection computation problem. By a simple distributed control rule, the considered multi-agent system with continuous-time dynamics achieves not only a consensus, but also an optimal agreement within the optimal solution set of the overall optimization objective. Directed and bidirectional communications are studied, respectively, and connectivity conditions are given to ensure a global optimal consensus. In this way, the corresponding intersection computation problem is solved by the proposed decentralized continuous-time algorithm. We establish several important properties of the distance functions with respect to the global optimal solution set and a class of invariant sets with the help of convex and non-smooth analysis.