A distributed algorithm for computing a common fixed point of a family of strongly quasi-nonexpansive maps

Abstract

This paper studies a distributed algorithm for finding a common fixed point of a family of m > 1 nonlinear maps Mi: ℝⁿ → ℝⁿ assuming that each map is strongly quasi-nonexpansive, and that at least one such common fixed point exists. A common fixed point is simultaneously and recursively computed by m agents assuming that each agent i knows only Mi, the current estimates of the fixed point generated by its neighbors, and nothing more. Neighbor relationships are described by a time-varying directed graph ℕ(t) whose vertices correspond to agents and whose arcs depict neighbor relationships. It is shown that for any sequence of repeatedly jointly strongly connected neighbor graphs ℕ(t), t ∈ {1, 2, …}, the algorithm causes all agents' estimates to converge to a common fixed point of Mi, i ∈ {1, 2, …, m}.