Improvement of a Distributed Algorithm for Solving Linear Equations

Wang, Xuan; Mou, Shaoshuai; Sun, Dong

doi:10.1109/tie.2016.2636119

Cited by 49 publications

(28 citation statements)

References 34 publications

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“…Basically, the algorithm with updating rule (7) is the plain generalization to the robust algorithm proposed in [17], which achieves the aforementioned condition by confining the injected errors in the kernel space of A i . In parallel with [17], we find that the work in [34] also developed an initialization-free algorithm by adding one additional term to (1). With appropriate expression manipulation, the approach proposed in [34] coincides with the updating rule (7).…”

Section: Robust Updating Rulementioning

confidence: 96%

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Securely Solving Linear Algebraic Equations in a Distributed Framework Enhanced With Communication-Efficient Algorithms

Yin

Shen

Cao

et al. 2020

IEEE Trans. Netw. Sci. Eng.

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Solving linear algebraic equations (a.k.a., an LAE problem) distributedly in a network with multiple agents has wide applications in distributed control, estimation and signal processing. A consensus-based distributed computing framework is studied in this paper. Specifically, each agent knows only a subproblem of the LAE, i.e., a subset of all equations, and then all agents apply a consensus-based algorithm to update their estimates of the correct solution of the LAE problem iteratively. Under certain conditions, it has been shown that all the estimates converge to the exact solution exponentially fast. However, such a distributed paradigm is vulnerable to malicious behaviors in an adversarial environment. In this paper, we indicate a number of security threats in this process, and thus develop robust computing solutions against those attacks. With particular attention to low storage overhead, we develop a new alternating projection method to enhance the consensus algorithm. Furthermore, we design an innovative misbehavior detection mechanism by exploiting the homomorphic signature technique. Our method can detect misbehaving agents without the common, yet sometimes infeasible, assumption of local good majority. Another significant contribution presented in this paper is an original component dropping approach for mitigating the communication overhead during the consensus process. From both theoretical and engineering perspectives, we study how the consensus can still be reached when a big portion of elements in exchanged message vectors are dropped. In our preliminary numerical results, roughly 80% data transmission can be reduced per epoch at the expense of 35% extra epochs to reach the consensus, implying 73% reduction in terms of communication overhead.

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Section: Robust Updating Rulementioning

confidence: 96%

“…In parallel with [17], we find that the work in [34] also developed an initialization-free algorithm by adding one additional term to (1). With appropriate expression manipulation, the approach proposed in [34] coincides with the updating rule (7).…”

Section: Robust Updating Rulementioning

confidence: 96%

Securely Solving Linear Algebraic Equations in a Distributed Framework Enhanced With Communication-Efficient Algorithms

Yin

Shen

Cao

et al. 2020

IEEE Trans. Netw. Sci. Eng.

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“…A significant amount of effort in the control community has recently been given to distributed algorithms for solving linear equations over multi-agent networks, in which each agent only knows part of the equation and controls a state vector that can be looked at as an estimate of the solution of the overall linear equations [1][2][3][4][5]. Numerous extensions along this direction include achieving solutions with the minimum Euclidean norm [6,7], elimination of the initialization step [8], reduction of state vector dimension by utilizing the sparsity of the linear equation [9] and achieving least square solutions [10][11][12][13][14][15]. All these algorithms yield asymptotic convergence, but require an infinite number of sensing or communication events.…”

Section: Introductionmentioning

confidence: 99%

Finite-Time Distributed Linear Equation Solver for Solutions With Minimum $l_1$-Norm

Zhou

Wang

Mou

et al. 2020

IEEE Trans. Automat. Contr.

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This paper proposes distributed algorithms for multi-agent networks to achieve a solution in finite time to a linear equation Ax = b where A has full row rank, and with the minimum l1-norm in the underdetermined case (where A has more columns than rows). The underlying network is assumed to be undirected and fixed, and an analytical proof is provided for the proposed algorithm to drive all agents' individual states to converge to a common value, viz a solution of Ax = b, which is the minimum l1norm solution in the underdetermined case. Numerical simulations are also provided as validation of the proposed algorithms.

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“…Merkezileştirilmiş algoritmaların aksine, dağıtık algoritmalar çok fazla değişken sayısına sahip doğrusal denklem sistemlerinin çözümü için iyi bir alternatif olması nedeniyle son 10 yılda araştırmacılar tarafından çalışılmaktadır [7,[9][10][11][12][13][14][15][16][17][18][19][20][21][22]. Bu dağıtık algoritmalar, denklem sisteminin yalnızca bir kısmını içeren birden fazla sayıda alt sisteme ayırmakta ve her alt sistemin çok etmenli sistemleri oluşturan bir etmen tarafından ele alındığı ve komşu etmenlerle bilgi paylaşımı yaparak tüm denklem sisteminin çözümünü bulmalarını sağlamaktadır.…”

Section: Introductionunclassified

Distributed Solution of Road Lighting Problem Over Multi-Agent Networks

Cihan

2020

Sakarya University Journal of Computer and Information Sciences

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In this study, we consider the solution of the road lighting problem by distributed algorithms over multi-agent networks where the objective is to determine the powers of the lamps that provide the desired road lighting level for a given road profile. The road is modeled as multiple road sections each with a length of 50 meters where a lighting pole is located in the middle of each section. Under given assumptions, the illumination levels of the road sections are expressed as linear functions of the powers of the lamps. When the processing units in the lighting poles can communicate wirelessly with the neighboring processing units and make simple calculations, it is shown that the power levels of the lamps that provide the desired lighting level for each road section can be calculated in a distributed manner. Finally, the model and the proposed solution has been verified by a numerical example.

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Improvement of a Distributed Algorithm for Solving Linear Equations

Cited by 49 publications

References 34 publications

Securely Solving Linear Algebraic Equations in a Distributed Framework Enhanced With Communication-Efficient Algorithms

Securely Solving Linear Algebraic Equations in a Distributed Framework Enhanced With Communication-Efficient Algorithms

Finite-Time Distributed Linear Equation Solver for Solutions With Minimum $l_1$-Norm

Distributed Solution of Road Lighting Problem Over Multi-Agent Networks

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