A Distributed Algorithm for Solving Linear Algebraic Equations Over Random Networks

Alaviani, Seyyed Shaho; Elia, Nicola

doi:10.1109/tac.2020.3010264

Cited by 26 publications

(14 citation statements)

References 40 publications

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“…Algorithm 1 Distributed Quasi-Averaged Operator Tracking (DOT) 1: Initialization: Stepsize α in (26), and local initial conditions x i,0 ∈ H and y i,0 = F i (x i,0 ) for all i ∈ [N ].…”

Section: The Dot Algorithm For Problem Imentioning

confidence: 99%

“…Meanwhile, a finite faimily of strongly quasi-nonexpansive operators were addressed in [18] for the common fixed point seeking problem. It should be noted that many interesting problems can boil down to the common fixed point finding problem, such as convex feasibility problems [22], [23] and the problem of solving linear algebraic equations in a distributed fashion [24]- [26], and so forth. Notice that all the aforementioned works are in the Euclidean space.…”

Section: Introductionmentioning

confidence: 99%

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DOT and DOP: Linearly Convergent Algorithms for Finding Fixed Points of Multi-Agent Operators

Li,

Meng,

Xie

2020

Preprint

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This paper investigates the distributed fixed point finding problem for a global operator over a directed and unbalanced multi-agent network, where the global operator is quasinonexpansive and only partially accessible to each individual agent. Two cases are addressed, that is, the global operator is sum separable and block separable. For this first case, the global operator is the sum of local operators, which are assumed to be Lipschitz, and each local operator is privately known to each individual agent. To deal with this scenario, a distributed (or decentralized) algorithm, called Distributed quasi-averaged Operator Tracking algorithm (DOT), is proposed and rigorously analyzed, and it is shown that the algorithm can converge to a fixed point of the global operator at a linear rate under a bounded linear regularity condition, which is strictly weaker than the strong convexity assumption on cost functions in existing convex optimization literature. For the second scenario, the global operator is composed of a group of local block operators, which are Lipschitz and can be accessed only by each individual agent. In this setup, a distributed algorithm, called Distributed quasiaveraged Operator Playing algorithm (DOP), is developed and shown to be linearly convergent to a fixed point of the global operator under the bounded linear regularity condition. The above studied problems provide a unified framework for many interesting problems. As examples, the proposed DOT and DOP are exploited to deal with distributed optimization, feasible point seeking for convex inequalities, and multi-player games under partial-decision information. Finally, numerical examples are presented to corroborate the theoretical results.

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Section: The Dot Algorithm For Problem Imentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

DOT and DOP: Linearly Convergent Algorithms for Finding Fixed Points of Multi-Agent Operators

Li,

Meng,

Xie

2020

Preprint

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“…Remark 6 Lots of existing works focus on solving (19) in a distributed manner, such as [1,30,40,41]. However, these works study the case where each agent only knows some complete rows of R and corresponding entries of r, while each agent here can know a sub-matrix of R as given in (21), thus providing a new perspective for solving (19) in a distributed manner.…”

Section: Solving Linear Algebraic Equationsmentioning

confidence: 99%

“…Also, the common fixed point finding problem for a family of strongly quasi-nonexpansive operators was addressed in [22]. The aforementioned common fixed point seeking problem can find numerous applications, such as, in convex feasibility problems [15,31] and solving linear algebraic equations in a distributed approach [1,30,[40][41][42], and so on. It should be noticed that the aforesaid works are only concerned with the Euclidean space.…”

Section: Introductionmentioning

confidence: 99%

Distributed Algorithms for Computing a Fixed Point of Multi-Agent Nonexpansive Operators

Xie

2019

Preprint

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This paper investigates the problem of finding a fixed point for a global nonexpansive operator under time-varying communication graphs in real Hilbert spaces, where the global operator is separable and composed of an aggregate sum of local nonexpansive operators. Each local operator is only privately accessible to each agent, and all agents constitute a network. To seek a fixed point of the global operator, it is indispensable for agents to exchange local information and update their solution cooperatively. To solve the problem, two algorithms are developed, called distributed Krasnosel'skiȋ-Mann (D-KM) and distributed block-coordinate Krasnosel'skiȋ-Mann (D-BKM) iterations, for which the D-BKM iteration is a block-coordinate version of the D-KM iteration in the sense of randomly choosing and computing only one block-coordinate of local operators at each time for each agent. It is shown that the proposed two algorithms can both converge weakly to a fixed point of the global operator. Meanwhile, the designed algorithms are applied to recover the classical distributed gradient descent (DGD) algorithm, devise a new block-coordinate DGD algorithm, handle a distributed shortest distance problem in the Hilbert space for the first time, and solve linear algebraic equations in a novel distributed approach. Finally, the theoretical results are corroborated by a few numerical examples.

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“…In the specific implementation, based on Gaussian belief propagation [19–21] and finite‐time consensus protocol [22–25], we propose our Generalized Belief Propagation algorithm. In order to compare the effect of this algorithm, based on the classic relaxation methods [26–28], we generalize the Generalized Gradient method for comparison.…”

Section: Introductionmentioning

confidence: 99%

Distributed convex optimization based on ADMM and belief propagation methods

Zhang

2020

Asian Journal of Control

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We consider a distributed convex optimization problem in which a connected network of agents collaboratively seeks to minimize the sum of their local objective functions over a common decision variable. We propose a new distributed optimization method in the Alternating Direction Method of Multipliers (ADMM) framework, But our method combines the celebrated Belief Propagation(BP) algorithm and relaxation iteration method to achieve distributed optimization. Numerical simulation shows that our proposed algorithms have good convergence speed. We also discuss the trade‐off between the convergence rate and the required communication and computational complexities.

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A Distributed Algorithm for Solving Linear Algebraic Equations Over Random Networks

Cited by 26 publications

References 40 publications

DOT and DOP: Linearly Convergent Algorithms for Finding Fixed Points of Multi-Agent Operators

DOT and DOP: Linearly Convergent Algorithms for Finding Fixed Points of Multi-Agent Operators

Distributed Algorithms for Computing a Fixed Point of Multi-Agent Nonexpansive Operators

Distributed convex optimization based on ADMM and belief propagation methods

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