Feedback-Feedforward Control Approach to Distributed Optimization

Zhirong Qiu, Lihua Xie, Keyou You

Abstract: In this work we study a basic distributed convex optimization problem where a network of nodes aims to reach consensus at the minimum of the sum of private convex costs. By treating the Laplacian operator under a weight-balanced and strongly connected network as a pseudo projector, we consider the problem from a feedback-feedforward control viewpoint, and design distributed algorithms by using proportional-integral-feedforward control techniques. With the aid of quadratic Lyapunov functions, we show the convergence of the proposed algorithms to a common optimum for L-smooth costs; if the cost is also strongly convex, then the convergence is exponentially fast. We further analyze and compare the convergence rate of proportional-integral (PI) and proportional-feedforward (PF) controllers for a class of quadratic costs, and show that the PF controller achieves better convergence with a proper choice of control gains.