Constrained Optimal Consensus in Dynamical Networks

TL;DR

This paper develops a distributed algorithm for multi-agent networks to reach an optimal consensus point that minimizes local convex functions while satisfying local inequality constraints, using primal-dual methods and consensus protocols.

Contribution

It introduces a novel integration of primal-dual gradient methods with consensus protocols for constrained optimization in dynamical networks.

Findings

Proves asymptotic convergence to the optimal point under certain conditions.

Demonstrates the effectiveness of the proposed protocol through a numerical example.

Ensures the consensus point satisfies KKT conditions for optimality.

Abstract

In this paper, an optimal consensus problem with local inequality constraints is studied for a network of single-integrator agents. The goal is that a group of single-integrator a gents rendezvous at the optimal point of the sum of local convex objective functions. The local objective functions are only available to the corresponding agents that only need to know their relative distances from their neighbors in order to seek the final optimal point. This point is supposed to be confined by some local inequality constraints. To tackle this problem, we integrate the primal dual gradient-based optimization algorithm with a consensus protocol to drive the agents toward the agreed point that satisfies KKT conditions. The asymptotic convergence of the solution of the optimization problem is proven with the help of LaSalle's invariance principle for hybrid systems. A numerical example is presented to show the effectiveness of our protocol.