Distributed Constrained Optimal Consensus under Fixed Time Delays

We study a distributed constrained optimal consensus problem of discrete-time multi-agent systems under fixed communication delays. Specifically, the total cost is expressed as the sum of individual cost of each agent, and only part of agents have access to the constraint. The constrained optimal consensus is solved if the final consensus value falls within the constraint, and in the meanwhile minimizes the total cost. Based on consensus method and subgradients, we propose a distributed two-step update scheme in which the state of each agent is firstly averaged with the delayed state information from neighbors, followed by a decaying subgradient descent from the individual cost, together with a movement along the projection direction if the agent can access the constraint. We show that the distributed constrained optimal consensus problem under fixed communication delays can be solved if the fixed network is balanced and contains a spanning tree, the constraint is accessible by at least one agent, and the gain on subgradient is decaying but persistent. Simulation results are provided to verify our conclusion.