On the O(1/k) Convergence of Asynchronous Distributed Alternating Direction Method of Multipliers

Wei, Ermin; Ozdaglar, Asuman

doi:10.48550/arxiv.1307.8254

Cited by 37 publications

(58 citation statements)

References 35 publications

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“…Furthermore, the required Lagrange multipliers from the other agents j ∈ M i are tracked by agent i using the auxiliary vector ξ i = col{ξ i j , j ∈ M i }, where each ξ i j is updated in (10). Additionally, each agent i must compute the interpolated point of ξ i j , denoted by ξi j in (11). This step is different than the steps in Algorithm 1, where the exchanged information is actually the interpolated point λi .…”

Section: Proposed Methodsmentioning

confidence: 99%

See 1 more Smart Citation

Accelerated Multi-Agent Optimization Method over Stochastic Networks

Ananduta¹,

Ocampo-Martinez²,

Nedić³

2020

Preprint

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We propose a distributed method to solve a multi-agent optimization problem with strongly convex cost function and equality coupling constraints. The method is based on Nesterov's accelerated gradient approach and works over stochastically time-varying communication networks. We consider the standard assumptions of Nesterov's method and show that the sequence of the expected dual values converge toward the optimal value with the rate of O(1/k 2 ). Furthermore, we provide a simulation study of solving an optimal power flow problem with a well-known benchmark case.

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Section: Proposed Methodsmentioning

confidence: 99%

“…where the second equality is obtained by using (11) and since λ i (k) = ξ i i (k), for any k ≥ 0, due to (10) and a proper initialization in Algorithm 2.…”

Section: B Proof Of Theoremmentioning

confidence: 99%

Accelerated Multi-Agent Optimization Method over Stochastic Networks

Ananduta¹,

Ocampo-Martinez²,

Nedić³

2020

Preprint

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show abstract

“…We model the communication network as a random graph [17]. To that end, let the communication network be described as an undirected graph G c (k) = (N , E c (k)), where E c (k) ⊆ E denotes the set of communication links that are active at iteration k−1, i.e., {i, j} ∈ E c (k) means that agents i and j can exchange information between each other.…”

Section: Time-varying Communication and Asynchronicitymentioning

confidence: 99%

Distributed Augmented Lagrangian Method for Link-Based Resource Sharing Problems of Multi-Agent Systems

Ananduta,

Nedić,

Ocampo-Martinez

2021

Preprint

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A multi-agent optimization problem motivated by the management of energy systems is discussed. The associated cost function is separable and convex although not necessarily strongly convex and there exist edge-based coupling equality constraints. In this regard, we propose a distributed algorithm based on solving the dual of the augmented problem. Furthermore, we consider that the communication network might be time-varying and the algorithm might be carried out asynchronously. The time-varying nature and the asynchronicity are modeled as random processes. Then, we show the convergence and the convergence rate of the proposed algorithm under the aforementioned conditions.

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“…However, despite these recent successes, it appears that the aforementioned works still pose very restrictive assumptions on the problem types in order to achieve convergence. For example it is not clear whether the ADMM can be used for the distributed optimization problem (3) over an arbitrary connected graph, despite the fact that for convex problem such application is popular and efficient [6,8,9,43,44].…”

Section: Literature Reviewmentioning

confidence: 99%

Decomposing Linearly Constrained Nonconvex Problems by a Proximal Primal Dual Approach: Algorithms, Convergence, and Applications

Hong

2016

Preprint

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In this paper, we propose a new decomposition approach named the proximal primal dual algorithm (Prox-PDA) for smooth nonconvex linearly constrained optimization problems. The proposed approach is primal-dual based, where the primal step minimizes certain approximation of the augmented Lagrangian of the problem, and the dual step performs an approximate dual ascent. The approximation used in the primal step is able to decompose the variable blocks, making it possible to obtain simple subproblems by leveraging the problem structures. Theoretically, we show that whenever the penalty parameter in the augmented Lagrangian is larger than a given threshold, the Prox-PDA converges to the set of stationary solutions, globally and in a sublinear manner (i.e., certain measure of stationarity decreases in the rate of O(1/r), where r is the iteration counter). Interestingly, when applying a variant of the Prox-PDA to the problem of distributed nonconvex optimization (over a connected undirected graph), the resulting algorithm coincides with the popular EXTRA algorithm [Shi et al 2014], which is only known to work in convex cases. Our analysis implies that EXTRA and its variants converge globally sublinearly to stationary solutions of certain nonconvex distributed optimization problem. There are many possible extensions of the Prox-PDA, and we present one particular extension to certain nonconvex distributed matrix factorization problem.

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On the O(1/k) Convergence of Asynchronous Distributed Alternating Direction Method of Multipliers

Cited by 37 publications

References 35 publications

Accelerated Multi-Agent Optimization Method over Stochastic Networks

Accelerated Multi-Agent Optimization Method over Stochastic Networks

Distributed Augmented Lagrangian Method for Link-Based Resource Sharing Problems of Multi-Agent Systems

Decomposing Linearly Constrained Nonconvex Problems by a Proximal Primal Dual Approach: Algorithms, Convergence, and Applications

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