Convergence rates for an inexact ADMM applied to separable convex optimization

• Published in: Computational optimization and applications Vol. 77; no. 3; pp. 729 - 754
• Main Authors: Hager, William W., Zhang, Hongchao
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.12.2020; Springer Nature B.V
• Subjects: Algorithms; Computational geometry; Convergence; Convex analysis; Convex and Discrete Geometry; Convexity; Ergodic processes; Iterative methods; Management Science; Mathematics; Mathematics and Statistics; Operations Research; Operations Research/Decision Theory; Optimization; Statistics; Alternating direction method of multipliers; Global convergence; Inexact methods; Accelerated gradient method; 90C25; Convergence rates; 90C06; 65Y20; Separable convex optimization; ADMM
• ISSN: 0926-6003; 1573-2894
• DOI: 10.1007/s10589-020-00221-y

Convergence rates are established for an inexact accelerated alternating direction method of multipliers (I-ADMM) for general separable convex optimization with a linear constraint. Both ergodic and non-ergodic iterates are analyzed. Relative to the iteration number k , the convergence rate is O ( 1 / k ) in a convex setting and O ( 1 / k 2 ) in a strongly convex setting. When an error bound condition holds, the algorithm is 2-step linearly convergent. The I-ADMM is designed so that the accuracy of the inexact iteration preserves the global convergence rates of the exact iteration, leading to better numerical performance in the test problems.