A proximal fully parallel splitting method with a relaxation factor for separable convex programming

In this paper, we propose a proximal fully parallel splitting method with a relaxation factor for solving separable convex minimization model with linear constraints, where the objective function is the sum of m individual functions without coupled variables. With a full Jacobian decomposition, we d...

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Bibliographic Details
Published in:Applied numerical mathematics Vol. 195; pp. 17 - 38
Main Authors: Yin, Jianghua, Jian, Jinbao, Jiang, Xianzhen, Wu, Jiansheng, Ma, Guodong
Format: Journal Article
Language:English
Published: Elsevier B.V 01.01.2024
Subjects:
Augmented Lagrangian method
Convergence rate
Relaxation factor
Splitting algorithm
Step size
Step size
Splitting algorithm
Convergence rate
Augmented Lagrangian method
Relaxation factor
ISSN:0168-9274
Online Access:Get full text
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Summary:In this paper, we propose a proximal fully parallel splitting method with a relaxation factor for solving separable convex minimization model with linear constraints, where the objective function is the sum of m individual functions without coupled variables. With a full Jacobian decomposition, we decompose the subproblem associated with the augmented Lagrangian method into m smaller subproblems and then add a quadratic proximal term to each decomposed subproblem, which makes the resulting ones easier to solve for many applications. In order to accelerate the numerical performance, we attach a positive relaxation factor to update the Lagrange multiplier, which also allows more flexibility in the design of algorithms. Moreover, we refine the step size of the underrelaxation step, which enlarges several existing ones in the literature. We prove that the proposed method is globally convergent, and show the worst-case O(1/k) convergence rate in a nonergodic sense. Finally, the efficiency and robustness of the proposed method are also demonstrated by solving the ℓ1 norm problem, the ℓ1-regularized least squares problem, the exchange problem and the total variation image restoration problem.
ISSN:0168-9274
DOI:10.1016/j.apnum.2023.09.003