scholarly journals A Proximal Alternating Direction Method of Multipliers with a Substitution Procedure

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Miantao Chao ◽  
Yongxin Zhao ◽  
Dongying Liang
Keyword(s):  
Linear Constraints ◽  
Alternating Direction Method ◽  
Convex Programming Problem ◽  
Method Of Multipliers ◽  
Worst Case ◽  
Alternating Direction ◽  
Separable Convex Programming ◽  
Contractive Type ◽  
Gradient Based ◽  
Substitution Procedure

In this paper, we considers the separable convex programming problem with linear constraints. Its objective function is the sum of m individual blocks with nonoverlapping variables and each block consists of two functions: one is smooth convex and the other one is convex. For the general case m≥3, we present a gradient-based alternating direction method of multipliers with a substitution. For the proposed algorithm, we prove its convergence via the analytic framework of contractive-type methods and derive a worst-case O1/t convergence rate in nonergodic sense. Finally, some preliminary numerical results are reported to support the efficiency of the proposed algorithm.

A Linearized Alternating Direction Method of Multipliers with Substitution Procedure

2015 ◽  
Vol 32 (03) ◽  
pp. 1550011 ◽  
Author(s):  
Miantao Chao ◽  
Caozong Cheng ◽  
Haibin Zhang
Keyword(s):  
Alternating Direction Method ◽  
Convex Programming Problem ◽  
Method Of Multipliers ◽  
Worst Case ◽  
Alternating Direction ◽  
Separable Convex Programming ◽  
Contractive Type ◽  
Linearly Constrained ◽  
Contraction Type ◽  
Substitution Procedure

We consider the linearly constrained separable convex programming problem whose objective function is separable into m individual convex functions with non-overlapping variables. The alternating direction method of multipliers (ADMM) has been well studied in the literature for the special case m = 2, but the direct extension of ADMM for the general case m ≥ 2 is not necessarily convergent. In this paper, we propose a new linearized ADMM-based contraction type algorithms for the general case m ≥ 2. For the proposed algorithm, we prove its convergence via the analytic framework of contractive type methods and we derive a worst-case O(1/t) convergence rate in ergodic sense. Finally, numerical results are reported to demonstrate the effectiveness of the proposed algorithm.

A Symmetric Alternating Direction Method of Multipliers for Separable Nonconvex Minimization Problems

2017 ◽  
Vol 34 (06) ◽  
pp. 1750030 ◽  
Author(s):  
Zhongming Wu ◽  
Min Li ◽  
David Z. W. Wang ◽  
Deren Han
Keyword(s):  
Linear Constraints ◽  
Bounded Sequence ◽  
Alternating Direction Method ◽  
Method Of Multipliers ◽  
Penalty Parameter ◽  
Augmented Lagrangian Function ◽  
Nonconvex Minimization ◽  
Nonconvex Functions ◽  
Alternating Direction ◽  
The Given

In this paper, we propose a symmetric alternating method of multipliers for minimizing the sum of two nonconvex functions with linear constraints, which contains the classic alternating direction method of multipliers in the algorithm framework. Based on the powerful Kurdyka–Łojasiewicz property, and under some assumptions about the penalty parameter and objective function, we prove that each bounded sequence generated by the proposed method globally converges to a critical point of the augmented Lagrangian function associated with the given problem. Moreover, we report some preliminary numerical results on solving [Formula: see text] regularized sparsity optimization and nonconvex feasibility problems to indicate the feasibility and effectiveness of the proposed method.

scholarly journals A Dynamical Alternating Direction Multiplier Method for Two-Block Optimization Problems

2021 ◽  
Author(s):  
Miantao Chao ◽  
Liqun Liu
Keyword(s):  
Dynamical System ◽  
Optimization Problems ◽  
Suitable Condition ◽  
Time Discretization ◽  
Alternating Direction Method ◽  
Lagrangian Function ◽  
Method Of Multipliers ◽  
Worst Case ◽  
Alternating Direction ◽  
Separable Optimization

Abstract In this paper, we propose a dynamic alternating direction method of multipliers for two-block separable optimization problems. The well-known classical ADMM can be obtained after the time discretization of the dynamical system. Under suitable condition, we prove that the trajectory asymptotically converges to a saddle point of the Lagrangian function of the problems. When the coefficient matrices in the constraint are identiy matrices, we prove the worst-case O(1/t) convergence rate in ergodic sense.

scholarly journals Convergence Analysis of Alternating Direction Method of Multipliers for a Class of Separable Convex Programming

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Zehui Jia ◽  
Ke Guo ◽  
Xingju Cai
Keyword(s):  
Convergence Analysis ◽  
Convex Programming ◽  
Alternating Direction Method ◽  
Linear Functions ◽  
Method Of Multipliers ◽  
Strongly Convex Functions ◽  
Strongly Convex ◽  
Alternating Direction ◽  
Separable Convex Programming ◽  
The Individual

The purpose of this paper is extending the convergence analysis of Han and Yuan (2012) for alternating direction method of multipliers (ADMM) from the strongly convex to a more general case. Under the assumption that the individual functions are composites of strongly convex functions and linear functions, we prove that the classical ADMM for separable convex programming with two blocks can be extended to the case with more than three blocks. The problems, although still very special, arise naturally from some important applications, for example, route-based traffic assignment problems.

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