Alternating direction method of multipliers for a class of nonconvex bilinear optimization: convergence analysis and applications

Hajinezhad, Davood; Shi, Qingjiang

doi:10.1007/s10898-017-0594-x

Cited by 30 publications

(21 citation statements)

References 33 publications

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“…Authors in [48] have presented a distributed ADMM for solving the direct current dynamic optimal power flow with carbon emission trading problem. In [49], Hajinezhad and Shi proposed an algorithm related to ADMM to study a class of nonconvex nonsmooth optimization problems with bilinear constraints which are widely used in machine learning and signal processing application domains.…”

Section: A General Literature Review On Distributed Methods For Solving Optimization Problemsmentioning

confidence: 99%

A Study on Distributed Optimization over Large-Scale Networked Systems

Abeynanda

Lanel

2021

Journal of Mathematics

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Distributed optimization is a very important concept with applications in control theory and many related fields, as it is high fault-tolerant and extremely scalable compared with centralized optimization. Centralized solution methods are not suitable for many application domains that consist of large number of networked systems. In general, these large-scale networked systems cooperatively find an optimal solution to a common global objective during the optimization process. Thus, it gives us an opportunity to analyze distributed optimization techniques that is demanded in most distributed optimization settings. This paper presents an analysis that provides an overview of decomposition methods as well as currently existing distributed methods and techniques that are employed in large-scale networked systems. A detailed analysis on gradient like methods, subgradient methods, and methods of multipliers including the alternating direction method of multipliers is presented. These methods are analyzed empirically by using numerical examples. Moreover, an example highlighting the fact that the gradient method fails to solve distributed problems in some circumstances is discussed under numerical results. A numerical implementation is used to demonstrate that the alternating direction method of multipliers can solve this particular problem, by revealing its robustness compared with the gradient method. Finally, we conclude the paper with possible future research directions.

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Section: A General Literature Review On Distributed Methods For Solving Optimization Problemsmentioning

confidence: 99%

A Study on Distributed Optimization over Large-Scale Networked Systems

Abeynanda

Lanel

2021

Journal of Mathematics

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“…It is ubiquitous nowadays in fields such as control [3][4][5], machine learning [6,7], signal and information processing [8,9], communication [10,11], and also NP-hard problems [12]. The existing research on multi-convex programming mainly solves some very special models [12][13][14][15][16][17][18][19]. These studies all give specific methods for each special model.…”

Section: Introductionmentioning

confidence: 99%

Partial Exactness for the Penalty Function of Biconvex Programming

Jiang

Shen

2021

Entropy

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Biconvex programming (or inequality constrained biconvex optimization) is an important model in solving many engineering optimization problems in areas like machine learning and signal and information processing. In this paper, the partial exactness of the partial optimum for the penalty function of biconvex programming is studied. The penalty function is partially exact if the partial Karush–Kuhn–Tucker (KKT) condition is true. The sufficient and necessary partially local stability condition used to determine whether the penalty function is partially exact for a partial optimum solution is also proven. Based on the penalty function, an algorithm is presented for finding a partial optimum solution to an inequality constrained biconvex optimization, and its convergence is proven under some conditions.

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“…Optimization of nonconvex functions is a much more challenging problem. Only recently, have there been developed a few nonconvex distributed optimization algorithms motivated by applications in resource allocation in ad-hoc network [17], sparse PCA [18], and flow control in communication networks [19]; see also [19][20][21][22][23][24][25][26][27] for additional algorithms developed for nonconvex optimization problems.…”

Section: Introductionmentioning

confidence: 99%

Gradient-Free Multi-Agent Nonconvex Nonsmooth Optimization

Hajinezhad

Zavlanos

2018

2018 IEEE Conference on Decision and Control (CDC)

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In this paper we consider the problem of minimizing the sum of nonconvex and possibly nonsmooth functions over a connected multi-agent network, where the agents have partial knowledge about the global cost function and can only access the zerothorder information (i.e., the functional values) of their local cost functions. We propose and analyze a distributed primal-dual gradient-free algorithm for this challenging problem. We show that by appropriately choosing the parameters, the proposed algorithm converges to the set of first order stationary solutions with a provable global sublinear convergence rate. Numerical experiments demonstrate the effectiveness of our proposed method for optimizing nonconvex and nonsmooth problems over a network.

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Alternating direction method of multipliers for a class of nonconvex bilinear optimization: convergence analysis and applications

Cited by 30 publications

References 33 publications

A Study on Distributed Optimization over Large-Scale Networked Systems

A Study on Distributed Optimization over Large-Scale Networked Systems

Partial Exactness for the Penalty Function of Biconvex Programming

Gradient-Free Multi-Agent Nonconvex Nonsmooth Optimization

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