Reaching an Optimal Consensus: Distributed Intersection Computation for Multi-agent Systems

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 achieves not only a consensus, but also an optimal one by agreeing within the optimal solution set of the group's 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 analysis and non-smooth analysis, based on which the convergence analysis is presented.