Reaching an Optimal Consensus: Dynamical Systems that Compute Intersections of Convex Sets
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.
Cite
@article{arxiv.1112.1333,
title = {Reaching an Optimal Consensus: Dynamical Systems that Compute Intersections of Convex Sets},
author = {Guodong Shi and Karl Henrik Johansson and Yiguang Hong},
journal= {arXiv preprint arXiv:1112.1333},
year = {2015}
}