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Revisiting Parallel Splitting Augmented Lagrangian Method: Tight Convergence and Ergodic Convergence Rate

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.

Author Details

Fan Jiang

Bingyan Lu

Zhongming Wu