Revisiting Parallel Splitting Augmented Lagrangian Method: Tight Convergence and Ergodic Convergence Rate
Year: 2024
Author: Fan Jiang, Bingyan Lu, Zhongming Wu
CSIAM Transactions on Applied Mathematics, Vol. 5 (2024), Iss. 4 : pp. 884–913
Abstract
This paper revisits the convergence and convergence rate of the parallel splitting augmented Lagrangian method, which can be used to efficiently solve the separable multi-block convex minimization problem with linear constraints. To make use of the separable structure, the augmented Lagrangian method with Jacobian-based decomposition fully exploits the properties of each function in the objective, and results in easier subproblems. The subproblems of the method can be solved and updated in parallel, thereby enhancing computational efficiency and speeding up the convergence. We further study the parallel splitting augmented Lagrangian method with a modified correction step, which shows improved performance with larger step sizes in the correction step. By introducing a refined correction step size with a tight bound for the constant step size, we establish the global convergence of the iterates and $\mathcal{O}(1/N)$ convergence rate in both the ergodic and non-ergodic senses for the new algorithm, where $N$ denotes the iteration numbers. Moreover, we demonstrate the applicability and promising efficiency of the method with tight step size through some applications in image processing.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/csiam-am.SO-2024-0013
CSIAM Transactions on Applied Mathematics, Vol. 5 (2024), Iss. 4 : pp. 884–913
Published online: 2024-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 30
Keywords: Convex programming augmented Lagrangian method optimal step size convergence rate.