Constrained Consensus and Optimization in Multi-Agent Networks

• Published in: IEEE transactions on automatic control Vol. 55; no. 4; pp. 922 - 938
• Main Authors: Nedic, A., Ozdaglar, A., Parrilo, P.A.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.04.2010; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Alignment; Applied sciences; Artificial intelligence; Automatic control; Autonomous agents; Computer science; control theory; systems; Computer systems and distributed systems. User interface; Consensus; Constraint optimization; Constraints; Convergence; Distributed algorithms; distributed optimization; Engineering profession; Estimates; Exact sciences and technology; Intelligent networks; Laboratories; Multiagent systems; Networks; Optimization; Software; subgradient algorithms; Systems engineering and theory; Averaging method; Convex set; distributed optimization; subgradient algorithms; Distributed system; Set constraint; constraints; Consensus; Constrained optimization; Projection method; Time varying system; Multiagent system; Optimal solution; Distributed algorithm; Convergence rate; Objective function; Artificial intelligence; Numerical convergence; Mathematical programming
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/TAC.2010.2041686

We present distributed algorithms that can be used by multiple agents to align their estimates with a particular value over a network with time-varying connectivity. Our framework is general in that this value can represent a consensus value among multiple agents or an optimal solution of an optimization problem, where the global objective function is a combination of local agent objective functions. Our main focus is on constrained problems where the estimates of each agent are restricted to lie in different convex sets. To highlight the effects of constraints, we first consider a constrained consensus problem and present a distributed "projected consensus algorithm" in which agents combine their local averaging operation with projection on their individual constraint sets. This algorithm can be viewed as a version of an alternating projection method with weights that are varying over time and across agents. We establish convergence and convergence rate results for the projected consensus algorithm. We next study a constrained optimization problem for optimizing the sum of local objective functions of the agents subject to the intersection of their local constraint sets. We present a distributed "projected subgradient algorithm" which involves each agent performing a local averaging operation, taking a subgradient step to minimize its own objective function, and projecting on its constraint set. We show that, with an appropriately selected stepsize rule, the agent estimates generated by this algorithm converge to the same optimal solution for the cases when the weights are constant and equal, and when the weights are time-varying but all agents have the same constraint set.