TY - JOUR
T1 - Revisiting Parallel Splitting Augmented Lagrangian Method: Tight Convergence and Ergodic Convergence Rate
AU - Jiang , Fan
AU - Lu , Bingyan
AU - Wu , Zhongming
JO - CSIAM Transactions on Applied Mathematics
VL - 4
SP - 884
EP - 913
PY - 2024
DA - 2024/11
SN - 5
DO - http://doi.org/10.4208/csiam-am.SO-2024-0013
UR - https://global-sci.org/intro/article_detail/csiam-am/23590.html
KW - Convex programming, augmented Lagrangian method, optimal step size, convergence rate.
AB - <p style="text-align: justify;">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.</p>