Continuous-Time Distributed Algorithm for Seeking Fixed Points of Multiagent Quasi-Nonexpansive Operators

This article investigates the distributed fixed point seeking problem for the operator in real Hilbert spaces over the multiagent networks, where each local operator can only be approximately and privately accessed by the corresponding agent. Specifically, two cases, including the global operator is sum separable and block separable, are considered. Under the assumption that the local operator is Lipschitz and the global operator is quasi-nonexpansive, continuous-time distributed algorithms over balanced time-varying digraphs and strongly connected fixed digraphs are designed for two cases, respectively, and it is proved in both cases that the algorithms weakly converge to a fixed point of the considered operator. To the best of our knowledge, these two algorithms are the first continuous-time dynamical system results for the distributed fixed point seeking problem. Meanwhile, the designed algorithms can also be applied to solve resource allocation, noncooperative games, and multicluster games, providing a unified framework for their solution. Finally, several numerical simulations are presented to verify the theoretical results.

I. Introduction

Fixed POINT theory is a powerful tool in several fields such as optimization, game theory, image recovery, inverse problems, and signal processing [1], providing a unified mathematical framework for the study of these problems. As a result, plenty of literature on the fixed point theory and its applications has emerged, especially in the study of the design of algorithms for computing the fixed point of operators.