Primal-Dual Fixed Point Algorithms Based on Adapted Metric for Distributed Optimization with Bounded Communication Delays

This paper concentrates on distributed optimization over networks with communication delays. Each subsystem in the network performs its local updates by using the information received from its neighbors, be it possibly outdated. The communication delays with respect to different neighbors are assumed to be arbitrary but bounded. The objective function consists of a twice differentiable coupling term and an aggregated private term. The private function of each subsystem is the sum of two possibly nonsmooth terms, one of which is composed of a linear mapping. We propose a primal-dual fixed point algorithm framework based on the adapted metric for two scenarios that the coupling among subsystems is only enacted by the global objective function and enforced both by the global objective function and the linear mapping. The adapted metric method utilizes an adequate quadratic approximation of the global objective function as the updating step-size to exploit the second-order information. Under some mild assumptions, the convergence of the proposed algorithms is rigorously analyzed based on the quasi-Fejér monotonicity. The numerical simulation verifies the correctness and effectiveness of the proposed algorithms.