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Accelerated Alternating Direction Method of Multipliers: an Optimal O(1/K) Nonergodic Analysis [article]

Huan Li, Zhouchen Lin
2018 arXiv   pre-print
Pasiliao, An Accelerated Linearized Alternating Direction Method of Multipliers, SIAM J. on Imaging Sciences, 2015, 1588-1623] and give an O(1/K) nonergodic convergence rate analysis, which satisfies |  ...  The Alternating Direction Method of Multipliers (ADMM) is widely used for linearly constrained convex problems.  ...  We modify the accelerated ADMM proposed in [18] and give an O(1/K) nonergodic analysis satisfying |F (x K ) − F (x * )| ≤ O(1/K) and Ax K − b ≤ O(1/K).  ... 
arXiv:1608.06366v5 fatcat:hgsmr7ghgzhz7bfvs2gkts2e5a

Faster and Non-ergodic O(1/K) Stochastic Alternating Direction Method of Multipliers [article]

Cong Fang, Feng Cheng, Zhouchen Lin
2017 arXiv   pre-print
Traditional Stochastic Alternating Direction Method of Multipliers and its Nesterov's acceleration scheme can only achieve ergodic O(1/√(K)) convergence rates, where K is the number of iteration.  ...  methods are actually tight O(1/√(K)) in non-ergodic sense.  ...  ACC-SADMM integrates Nesterov's extrapolation and VR techniques and achieves a non-ergodic O(1/K) convergence rate. We do experiments to demonstrate that our algorithm is faster than other SADMM.  ... 
arXiv:1704.06793v1 fatcat:dxpk4lvvovftlpe6obf7t32bkq

The Linear Convergence Analysis of The Linearized Version of The Generalized Alternating Direction Method of Multipliers for Convex Optimization Problems [article]

Jianwen Peng, Dexi Liu, Xueqing Zhang
2022 arXiv   pre-print
efficient and simple acceleration scheme of the aternating direction method of multipliers.  ...  To solve the separable convex optimization problem with linear constraints, Eckstein and Bertsekas introduced the generalized alternating direction method of multipliers (in short, GADMM), which is an  ...  The first author was supported by the National Natural Science Foundation of China (11991024), the Basic and Advanced Research Project of Chongqing (cstc2021jcyj-msxmX0300), the Team Project of Innovation  ... 
arXiv:2202.09610v4 fatcat:n4zmsn2dgnh6dnejwehbacqwza

Generalized alternating direction method of multipliers: new theoretical insights and applications

Ethan X. Fang, Bingsheng He, Han Liu, Xiaoming Yuan
2015 Mathematical Programming Computation  
Recently, the alternating direction method of multipliers (ADMM) has received intensive attention from a broad spectrum of areas.  ...  Theoretically, we show the worst-case O(1/k) convergence rate measured by the iteration complexity (k represents the iteration counter) in both the ergodic and a nonergodic senses for the linearized version  ...  Conclusion In this paper, we take a deeper look at the linearized version of the generalized alternating direction method of multiplier (ADMM) and establish its worstcase O(1/k) convergence rate in both  ... 
doi:10.1007/s12532-015-0078-2 pmid:28428830 pmcid:PMC5394583 fatcat:e5nocwp6v5gubcn6rg53pz5oky

Iterative Complexity and Its Applications of Linearized Generalized Alternating Direction Method of Multipliers with Multi-block Case [article]

He Jian and Zhang Bangzhong and Li Jinlin
2022 arXiv   pre-print
The linearized version of the generalized alternating direction method of multipliers (L-GADMM) is particularly efficient for the two-block separable convex programming problem and its convergence was  ...  Theoretically, we prove global convergence of the new method and establish the worst-case convergence rate in the ergodic and nonergodic senses for the proposed algorithm.  ...  These methods involve the Peaceman-Rachford splitting method (PRSM) [5, 16] , the alternating direction method of multipliers (ADMM) [14, 15] , and their variants [17] [18] [19] [20] [21] [22] [23]  ... 
arXiv:2204.08273v1 fatcat:rlqtqszldjekzmlentfqjmg4ly

Convergence Rate Analysis for the Alternating Direction Method of Multipliers with a Substitution Procedure for Separable Convex Programming

Bingsheng He, Min Tao, Xiaoming Yuan
2017 Mathematics of Operations Research  
Recently, in [17] we have showed the first possibility of combining the Douglas-Rachford alternating direction method of multipliers (ADMM) with a Gaussian back substitution procedure for solving a convex  ...  Without such an error bound assumption, we can still estimate the worst-case iteration complexity for this framework in both ergodic and nonergodic senses.  ...  A fundamental method in this regard is the Douglas-Rachford alternating direction method of multipliers (ADMM for short) proposed in [14] (see also [11] ).  ... 
doi:10.1287/moor.2016.0822 fatcat:as2gersjdngxbg7xhg4dpa4bcy

Block-wise Alternating Direction Method of Multipliers for Multiple-block Convex Programming and Beyond

Bingsheng He, Xiaoming Yuan
2015 SMAI Journal of Computational Mathematics  
The alternating direction method of multipliers (ADMM) is a benchmark for solving a linearly constrained convex minimization model with a two-block separable objective function; and it has been shown that  ...  its direct extension to a multiple-block case where the objective function is the sum of more than two functions is not necessarily convergent.  ...  For the special case of (1.1) with m = 2, the alternating direction method of multipliers (ADMM) in [14] reads as          x k+1 1 = arg min L 2 β (x 1 , x k 2 , λ k ) x 1 ∈ X 1 , x k+1 2 =  ... 
doi:10.5802/smai-jcm.6 fatcat:abv6ru27hnd67jor6ndlnawgue

Iteration complexity analysis of a partial LQP-based alternating direction method of multipliers [article]

Jianchao Bai, Yuxue Ma, Hao Sun, Miao Zhang
2021 arXiv   pre-print
By adding the Logarithmic Quadratic Proximal (LQP) regularizer with suitable proximal parameter to each of the first grouped subproblems, we develop a partial LQP-based Alternating Direction Method of  ...  Using a prediction-correction approach to analyze properties of the iterates generated by ADMM-LQP, we establish its global convergence and sublinear convergence rate of O(1/T) in the new ergodic and nonergodic  ...  An effective approach to overcome the above disadvantages is the Alternating Direction Method of Multipliers (ADMM) which could be regarded as a splitting version of ALM:                 ... 
arXiv:2103.16752v1 fatcat:bsdanhv3lbbplbhrqjvulhbjqy

Accelerated Proximal Point Method for Maximally Monotone Operators [article]

Donghwan Kim
2021 arXiv   pre-print
The proximal point method includes various well-known convex optimization methods, such as the proximal method of multipliers and the alternating direction method of multipliers, and thus the proposed  ...  This paper proposes an accelerated proximal point method for maximally monotone operators. The proof is computer-assisted via the performance estimation problem approach.  ...  The form of the resulting optimized method differs from that of the accelerated proximal point method proposed in this paper, but [62] recently showed that they are equivalent in the sense that they  ... 
arXiv:1905.05149v4 fatcat:3mefhskqvfbjxp4uw7k43x5nfa

1 Introduction [chapter]

2019 Non-Extensive Entropy Econometrics for Low Frequency Series  
The procedure remains in line with multiplier-accelerator analysis, assuming that induced investment is a function of expected growth.  ...  For this purpose, we propose using a dynamic model24 of I-O in which investment is the endogenous variable in the context of accelerator analysis of macroeconomic theory.  ...  Additional and consistent a priori information is required to enable an optimization entropy device moving from uniform distribution toward an optimal, global solution.  ... 
doi:10.1515/9783110605914-019 fatcat:yir426vzmjfgxpnhn2rmyjf6gq

Accelerated primal-dual methods for linearly constrained convex optimization problems [article]

Hao Luo
2022 arXiv   pre-print
This work proposes an accelerated primal-dual dynamical system for affine constrained convex optimization and presents a class of primal-dual methods with nonergodic convergence rates.  ...  sequentially and lead to new accelerated primal-dual methods for solving linearly constrained optimization problems.  ...  In this setting, or even more general multiblock case (cf. (48) ), the alternating direction method of multiplies (ADMM) is one of the most prevailing splitting algorithms.  ... 
arXiv:2109.12604v2 fatcat:5vsvqinhwjhm5cw3wqyxcyyccq

ADMM-based Distributed State Estimation for Power Systems: Evaluation of Performance [article]

Samal Kubentayeva, Elena Gryazina, Sergei Parsegov, Alexander Gasnikov, Federico Ibáñez
2020 arXiv   pre-print
We also thoroughly analyze the theoretical and practical performance, concluding that accelerated approach outperforms the existing ones.  ...  In this paper, we propose some novel approaches for speeding up the ADMM-based distributed state estimation algorithms by utilizing some recent results in optimization theory.  ...  Their approach is based on the alternating direction method of multipliers (ADMM).  ... 
arXiv:1911.11080v3 fatcat:wa4z2pelcrhn5f3ftoltp2ubje

Cumulative Author Index, Volumes 1–20

2009 Journal of the American Society for Mass Spectrometry  
.: Char-acterization of Long-Chain Carboxylic Acid Esters with CH 3 OBOCH 3 ϩ in a Small Fourier-Transform Ion Cyclotron Resonance Mass Spectrometry, 7:1138 Tholey, A. See Selevsek, N.  ...  -F., Turk, J.: Differentiation of 1-O-alk-1Ј-enyl-2-acyl and 1-O-alkyl-2-acyl Glycerophospholipids by Multiple-Stage Linear Ion-Trap Mass Spectrometry with Electrospray Ionization, 18:2065 Hsu, F.  ...  Jr.: Direct Observation of Trapping Motion in Elongated Fourier-Transform Mass Spectrometry Trapped Ion Cells, 1:351 Hofstadler, S.A., Laude, D.A.  ... 
doi:10.1016/s1044-0305(09)00877-0 fatcat:inwqpry4trci5eucc6y5ri3h2m

A primal-dual algorithm framework for convex saddle-point optimization

Benxin Zhang, Zhibin Zhu
2017 Journal of Inequalities and Applications  
The convergence rate O(1/t) in the ergodic and nonergodic senses is also given, where t denotes the iteration number.  ...  We prove the convergence of the proposed frame from the perspective of proximal point algorithm-like contraction methods and variational inequalities approach.  ...  Acknowledgements This work is supported by the National Natural Science Foundation of China (11361018, 11461015), Guangxi Natural Science Foundation (2014GXNSFFA118001), Guangxi Key Laboratory of Cryptography  ... 
doi:10.1186/s13660-017-1548-z pmid:29104405 pmcid:PMC5656743 fatcat:egolb76m6redjklwqecesh6lj4

A Strictly Contractive Peaceman--Rachford Splitting Method for Convex Programming

Bingsheng He, Han Liu, Zhaoran Wang, Xiaoming Yuan
2014 SIAM Journal on Optimization  
Compared to the Douglas-Rachford splitting method (DRSM), another splitting method from which the alternating direction method of multipliers originates, PRSM requires more restrictive assumptions to ensure  ...  A worst-case O(1/t) convergence rate of this strictly contractive PRSM in a nonergodic sense is established.  ...  The strictly contractive PRSM is as easy to implement as that of the alternating direction method of multipliers (ADMM), and it is numerically faster.  ... 
doi:10.1137/13090849x pmid:25620862 pmcid:PMC4302964 fatcat:xcdfh4czqjgwjir2paxu4mjbt4
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