Distributed Banach-Picard Iteration for Locally Contractive Maps

作     者：Andrade, Francisco Figueiredo, Mário A.T. Xavier, João 

作者机构：Instituto Superior Técnico Universidade de Lisboa Lisbon Portugal Instituto de Telecomunicações Lisbon Portugal Laboratory for Robotics and Engineering Systems Institute for Systems and Robotics Lisbon Portugal 

出 版 物：《arXiv》 (arXiv)

年 卷 期：2021年

核心收录：

主　　题：Mathematical operators 

摘      要：The Banach-Picard iteration is widely used to find fixed points of locally contractive (LC) maps. This paper extends the Banach-Picard iteration to distributed settings;specifically, we assume the map of which the fixed point is sought to be the average of individual (not necessarily LC) maps held by a set of agents linked by a communication network. An additional difficulty is that the LC map is not assumed to come from an underlying optimization problem, which prevents exploiting strong global properties such as convexity or Lipschitzianity. Yet, we propose a distributed algorithm and prove its convergence, in fact showing that it maintains the linear rate of the standard Banach-Picard iteration for the average LC map. As another contribution, our proof imports tools from perturbation theory of linear operators, which, to the best of our knowledge, had not been used before in the theory of distributed *** Codes 37C25 © 2021, CC BY-NC-ND.