Convergence of alternating direction method for minimizing sum of two nonconvex functions with linear constraints

2016 ◽  
Vol 94 (8) ◽  
pp. 1653-1669 ◽  
Author(s):  
K. Guo ◽  
D. R. Han ◽  
T. T. Wu
Keyword(s):  
Linear Constraints ◽  
Alternating Direction Method ◽  
Nonconvex Functions ◽  
Alternating Direction

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 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.

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