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

Abstract

A distributed algorithm is described for finding a common fixed point of a family of m > 1 nonlinear maps M i : R n → R n assuming that each map is a paracontraction and that at least one such 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 a fixed point by utilizing the current estimates generated by each of its neighbors. Neighbor relations are characterized by a time-varying directed graph N(t). It is shown under suitably general conditions on N(t), that the algorithm causes all agent estimates to converge to the same common fixed point of the m nonlinear maps.

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

“…where the inequality has employed Cauchy-Schwarz inequality. It remains to show the convergence rate (12). Let us prove it by contradiction.…”

“…It is apparent that (47) contradicts (37). Therefore, one can claim that (12) holds. This ends the proof of Theorem 1.…”

“…Let us prove it by contradiction. If there are no subsequences such that (12) holds, then there must exist k 0 ∈ N, C > 0, and i 0 ∈ [N ], such that for all k ≥ k 0…”

Multi-agent network games with applications in smart electric mobility

“…Furthermore, there exists k ∈ N such that, for all k > k, the matrix A(k) satisfies inf k>k min i∈N [A(k)] ii =: a > 0. We note that when the mappings proxf i 's have a common fixed point 1 , i.e., ∩ i∈N fix(proxf i ) = ∅, then our convergence problem boils down to the setup studied in [89]. In this case, [89, Th.…”

“…In [47] and Chapter 2, the condition on the double stochasticity of the adjacency matrix was relaxed, in the first case by means of a dwell time. Notice that these types of games can also be rephrased as paracontracions; in this framework, the work in [89] provided convergence for repeatedly jointly connected digraphs. Iterative equilibrium seeking algorithms were developed for constrained multi-agent network games in [99] under the assumption of a static communication network.…”