A Bregman-style Partially Symmetric Alternating Direction Method of Multipliers for Nonconvex Multi-block Optimization

• Published in: Acta Mathematicae Applicatae Sinica Vol. 39; no. 2; pp. 354 - 380
• Main Authors: Liu, Peng-jie, Jian, Jin-bao, Ma, Guo-dong
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.04.2023; Springer Nature B.V
• Edition: English series
• Subjects: Applications of Mathematics; Convergence; Iterative methods; Lagrange multiplier; Math Applications in Computer Science; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Optimization; Theoretical; Kurdyka-Łojasiewicz property; 90C30; 90C26; nonconvex optimization; convergence rate; alternating direction method with multipliers; multi-block optimization
• ISSN: 0168-9673; 1618-3932
• DOI: 10.1007/s10255-023-1048-5

The alternating direction method of multipliers (ADMM) is one of the most successful and powerful methods for separable minimization optimization. Based on the idea of symmetric ADMM in two-block optimization, we add an updating formula for the Lagrange multiplier without restricting its position for multi-block one. Then, combining with the Bregman distance, in this work, a Bregman-style partially symmetric ADMM is presented for nonconvex multi-block optimization with linear constraints, and the Lagrange multiplier is updated twice with different relaxation factors in the iteration scheme. Under the suitable conditions, the global convergence, strong convergence and convergence rate of the presented method are analyzed and obtained. Finally, some preliminary numerical results are reported to support the correctness of the theoretical assertions, and these show that the presented method is numerically effective.