A Distributed Algorithm for Computing a Common Fixed Point of a Family of Paracontractions

TL;DR

This paper introduces a distributed algorithm enabling multiple agents to collaboratively find a common fixed point of nonlinear maps called paracontractions, using only local neighbor information and ensuring convergence under certain graph conditions.

Contribution

It proposes a novel distributed method for computing common fixed points of paracontractions with convergence guarantees over time-varying directed graphs.

Findings

All agent estimates converge to a common fixed point.

The algorithm works for any family of paracontractions with at least one common fixed point.

Convergence is guaranteed under strongly connected neighbor graphs.

Abstract

A distributed algorithm is described for finding a common fixed point of a family of $m>1$ nonlinear maps $M_i : \mathbb{R}^n \rightarrow \mathbb{R}^n$ assuming that each map is a paracontraction and that such a common fixed point exists. The common fixed point is simultaneously computed by $m$ agents assuming each agent $i$ knows only $M_i$, the current estimates of the fixed point generated by its neighbors, and nothing more. Each agent recursively updates its estimate of the fixed point by utilizing the current estimates generated by each of its neighbors. Neighbor relations are characterized by a time-dependent directed graph $\mathbb{N}(t)$ whose vertices correspond to agents and whose arcs depict neighbor relations. It is shown that for any family of paracontractions $M_i, i \in \{1,2,\ldots,m\}$ which has at least one common fixed point, and any sequence of strongly connected neighbor graphs $\mathbb{N}(t)$, $t=1,2,\ldots$, the algorithm causes all agent estimates to converge to a common fixed point.