A Linearized Alternating Direction Method of Multipliers with Substitution Procedure

• Published in: Asia-Pacific journal of operational research Vol. 32; no. 3; p. 1550011
• Main Authors: Chao, Miantao, Cheng, Caozong, Zhang, Haibin
• Format: Journal Article
• Language: English
• Published: Singapore World Scientific Publishing Co. & Operational Research Society of Singapore 01.06.2015; World Scientific Publishing Co. Pte., Ltd
• Subjects: Algorithms; Computational geometry; Convergence; Convexity; Economic models; Linearization; Mathematical programming; Multipliers; Operations research; Alternating direction method of multipliers; convex programming; contraction methods; convergence rate
• ISSN: 0217-5959; 1793-7019; 0217-5959
• DOI: 10.1142/S0217595915500116

We consider the linearly constrained separable convex programming problem whose objective function is separable into m individual convex functions with non-overlapping variables. The alternating direction method of multipliers (ADMM) has been well studied in the literature for the special case m = 2, but the direct extension of ADMM for the general case m ≥ 2 is not necessarily convergent. In this paper, we propose a new linearized ADMM-based contraction type algorithms for the general case m ≥ 2. For the proposed algorithm, we prove its convergence via the analytic framework of contractive type methods and we derive a worst-case O(1/t) convergence rate in ergodic sense. Finally, numerical results are reported to demonstrate the effectiveness of the proposed algorithm.