A Distributed Algorithm for Solving Linear Algebraic Equations Over Random Networks

Alaviani, S. Sh.; Elia, N.

doi:10.48550/arxiv.1809.07955

Cited by 1 publication

(1 citation statement)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…operators was studied in [12], [13], which is motivated by a typical problem, that is, solving a linear algebraic equation in the Euclidean space in a distributed manner, where a multiple of agents hold private partial information on the linear equation and thus all agents need to cooperatively solve the problem through local communications [14]- [18]. Meanwhile, the case with strongly quasi-nonexpansive operators was reported in [19].…”

Section: Introductionmentioning

confidence: 99%

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

Liu

Fullmer

Nedić

et al. 2017

2017 American Control Conference (ACC)

View full text Add to dashboard Cite

This paper addresses the problem of seeking a common fixed point for a collection of nonexpansive operators over time-varying multi-agent networks in real Hilbert spaces, where each operator is only privately and approximately known to each individual agent, and all agents need to cooperate to solve this problem by propagating their own information to their neighbors through local communications over time-varying networks. To handle this problem, inspired by the centralized inexact Krasnosel'skiȋ-Mann (IKM) iteration, we propose a distributed algorithm, called distributed inexact Krasnosel'skiȋ-Mann (D-IKM) iteration. It is shown that the D-IKM iteration can converge weakly to a common fixed point of the family of nonexpansive operators. Moreover, under the assumption that all operators and their own fixed point sets are (boundedly) linearly regular, it is proved that the D-IKM iteration converges with a rate O(1/k ln(1/ξ) ) for some constant ξ ∈ (0, 1), where k is the iteration number. To reduce computational complexity and burden of storage and transmission, a scenario, where only a random part of coordinates for each agent is updated at each iteration, is further considered, and a corresponding algorithm, named distributed inexact block-coordinate Krasnosel'skiȋ-Mann (D-IBKM) iteration, is developed. The algorithm is proved to be weakly convergent to a common fixed point of the group of considered operators, and, with the extra assumption of (bounded) linear regularity, it is convergent with a rate O(1/k ln(1/ξ) ). Furthermore, it is shown that the convergence rate O(1/k ln(1/ξ) ) can still be guaranteed under a more relaxed (bounded) power regularity condition.Index Terms-Distributed algorithms, multi-agent networks, Krasnosel'skiȋ-Mann iteration, nonexpansive operators, fixed point, optimization.weakly to a fixed point of T under mild conditions [21], [25], [36], for example, when ∞ j=1 α j (1 − α j ) = ∞ for the KM iteration [36]. Now, let us introduce the graph theory for describing the communication pattern among all agents [11]. Specifically, the communication mode among N agents can be modeled by a digraph G = (V, E), where V = [N ] is the node or vertex set, and E ⊂ V × V is the edge set. An edge (i, j) ∈ E means that agent i is capable of transmitting its information to agent j, in which case agent i is called a neighbor of agent j. A directed path from i 1 to i l is a sequence of edges of the form (i 1 , i 2 ), (i 3 , i 4 ), . . . , (i l−1 , i l ). A graph is called strongly connected if there exists at least a directed path from any node to any other node in this graph. In this paper, the communication graph for all agents is assumed to be time-varying, that is, any two agents can have different communication status at different time steps. In this case, the graph is denoted asAt each time k ∈ N, there exists an adjacency matrix A k = (a ij,k ) ∈ R N ×N such that a ij,k > 0 if (j, i) ∈ E k , and a ij,k = 0 otherwise. Assume that a ii,k > 0 for all i ∈ [N ] and all k ∈ N. For communication graphs, we...

show abstract