Linear convergence rate for distributed optimization with the alternating direction method of multipliers

2014 ◽  
Author(s):  
F. Iutzeler ◽  
P. Bianchi ◽  
Ph. Ciblat ◽  
W. Hachem
Keyword(s):  
Convergence Rate ◽  
Distributed Optimization ◽  
Linear Convergence ◽  
Alternating Direction Method ◽  
Method Of Multipliers ◽  
Linear Convergence Rate ◽  
Alternating Direction

On the Convergence Rate of Inexact Majorized sGS ADMM with Indefinite Proximal Terms for Convex Composite Programming

2020 ◽  
pp. 2050035
Author(s):  
Min Li ◽  
Zhongming Wu
Keyword(s):  
Convergence Rate ◽  
Optimization Problem ◽  
Linear Convergence ◽  
Specific Form ◽  
Alternating Direction Method ◽  
Linear Convergence Rate ◽  
New Methods ◽  
Alternating Direction ◽  
Separable Optimization ◽  
Composite Programming

In this paper, we propose an inexact majorized symmetric Gauss–Seidel (sGS) alternating direction method of multipliers (ADMM) with indefinite proximal terms for multi-block convex composite programming. This method is a specific form of the inexact majorized ADMM which is further proposed to solve a general two-block separable optimization problem. The new methods adopt certain relative error criteria to solve the involving subproblems approximately, and the step-sizes allow to choose in the scope [Formula: see text]. Under more general conditions, we establish the global convergence and Q-linear convergence rate of the proposed methods.

scholarly journals A Study on Distributed Optimization over Large-Scale Networked Systems

2021 ◽  
Vol 2021 ◽  
pp. 1-19
Author(s):  
Hansi K. Abeynanda ◽  
G. H. J. Lanel
Keyword(s):  
Gradient Method ◽  
Large Scale ◽  
Distributed Optimization ◽  
Alternating Direction Method ◽  
Optimization Techniques ◽  
Networked Systems ◽  
Future Research ◽  
Subgradient Methods ◽  
Method Of Multipliers ◽  
Alternating Direction

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.

Image denoising algorithm of compressed sensing based on alternating direction method of multipliers

2021 ◽  
pp. 2250009
Author(s):  
Changjie Fang ◽  
Jingyu Chen ◽  
Shenglan Chen
Keyword(s):  
Compressed Sensing ◽  
Convergence Rate ◽  
Objective Function ◽  
Image Denoising ◽  
Alternating Direction Method ◽  
Method Of Multipliers ◽  
Alternating Direction ◽  
Time Simulation ◽  
Optimal Value ◽  
Simulation Results

In this paper, we propose an image denoising algorithm for compressed sensing based on alternating direction method of multipliers (ADMM). We prove that the objective function of the iterates approaches the optimal value. We also prove the [Formula: see text] convergence rate of our algorithm in the ergodic sense. At the same time, simulation results show that our algorithm is more efficient in image denoising compared with existing methods.

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