Distributed Banach–Picard Iteration for Locally Contractive Maps

The Banach–Picard iteration is widely used to find fixed points of locally contractive (LC) maps. This article extends the distributed settings; specifically, we assume map which point sought be average individual (not necessarily LC) maps held by a set agents linked communication network. An additional difficulty that LC not assumed come from an underlying optimization problem, prevents exploiting strong global properties, such as convexity or Lipschitzianity. Yet, propose algorithm and prove its convergence, in fact showing it maintains linear rate standard for map. As another contribution, our proof imports tools perturbation theory operators, which, best knowledge, are scarcely exploited computation.