Some convergence results for asynchronous algorithms

Abstract

Resum+.Nous pr6sentons dans cet article des r6sultats de convergence des algorithmes asynchrones bas6s essentiellement sur la notion classique de contraction.Nous g6n6ralisons, en particulier, tousles r6sultats de convergence de ces algorithmes qui font l'hypoth6se de contraction en norme vectorielle qui r6cemment a 6t6 tr6s souvant utilis6e.Par ailleurs, l'hypoth6se de contraction en norme vectorielle peut se trouver difficile, voire impossible /t v6rifier pour certains probl6mes qui peuvent ~tre cependant abordds dans le cadre de la contraction classique que nous adoptons. Summary.In this paper we present convergence results for the asynchronous algorithms based essentially on the notion of classical contraction.We generalize, in particular, all convergence results for those algorithms which are based on the vectorial norm hypothesis, in wide spread use recently.Certain problems, for which the vectorial norm hypothesis can be difficult or even impossible to verify, can nontheless be tackled within the scope of the classical contraction that we adopte.Subject classifications. AM S(MOS): 65H 10; C R: 5.15.

On the convergence rates of asynchronous iterations

“…Some of the first results on the convergence of asynchronous iterations were derived by Chazan and Miranker [1], who studied chaotic relaxations for solving linear systems of equations. Several authors have proposed extensions of this pioneering work to nonlinear iterations involving maximum norm contractions (e.g., [2], [3]) and for monotone iterations (e.g., [4], [5]). Powerful convergence results for broad classes of asynchronous algorithms, including maximum norm contractions and monotone mappings, under different assumptions on communication delays and update rates were presented by Bertsekas [6] and Bertsekas and Tsitsiklis [7].…”

An Agda Formalization of Üresin & Dubois’ Asynchronous Fixed-Point Theory

“…For instance El Tarazi [8] shows that if there is a normed linear space over each the values at each node i, then convergence occurs if there exists a fixed point x * and a γ ∈ (0, 1] such that:…”

Parallel Syncrhronous Algorithm for Nonlinear Fixed Point Problems

“…For more details about asynchronous algorithms see [1], [2], [3] and [4]. In the following theorem, Bahi [3] has shown the convergence of the sequence {x p } defined by (2.1) in the synchronous linear case i.e s i (p) = p, ∀p ∈ {1, ..., α} and F is a linear operator.…”