Distributed Optimization Algorithm Design and Analysis on Cooperation-Competition Network Based on PID Control

This paper addresses distributed optimization problems with equality constraints within the framework of cooperation-competition network. Motivated by the exceptional performance of proportional-integral-derivative (PID) controllers, we propose an accelerated distributed optimization algorithm to decompose the distributed optimization problem on the cooperation-competition network into two cooperative subnetworks and address them. By leveraging the Lyapunov stability theorem, we establish the exponential convergence of our algorithm over undirected connected and structurally balanced cooperation-competition graphs when the parameters (e.g., kp, ki, kd) are selected within certain ranges, assuming that the local objective functions are smooth and strongly convex. Additionally, we provide guidelines for selecting appropriate parameter values (e.g., kp, ki, kd). Furthermore, we show that the D-PID-CCN has great potential for nonconvex distributed optimization over the cooperation-competition network. Finally, we present the effectiveness and superiority of our proposed algorithms on several numerical simulations.