Continuous-time Proportional-Integral Distributed Optimization for Networked Systems

School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA. Graduate School of Informatics, Kyoto University, Kyoto 6068501, Japan. Abstract In this paper we explore the relationship between dual decomposition and the consensus-based method for distributed optimization. The relationship is developed by examining the similarities between the two approaches and their relationship to gradient-based constrained optimization. By formulating each algorithm in continuous-time, it is seen that both approaches use a gradient method for optimization with one using a proportional control term and the other using an integral control term to drive the system to the constraint set. Therefore, a significant contribution of this paper is to combine these methods to develop a continuous-time proportional-integral distributed optimization method. Furthermore, we establish convergence using Lyapunov stability techniques and utilizing properties from the network structure of the multi-agent system.