Optimally linearizing the alternating direction method of multipliers for convex programming

Bingsheng He (), Feng Ma () and Xiaoming Yuan ()
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Bingsheng He: Southern University of Science and Technology of China
Feng Ma: High-Tech Institute of Xi’an
Xiaoming Yuan: The University of Hong Kong

Computational Optimization and Applications, 2020, vol. 75, issue 2, No 2, 388 pages

Abstract: Abstract The alternating direction method of multipliers (ADMM) is being widely used in a variety of areas; its different variants tailored for different application scenarios have also been deeply researched in the literature. Among them, the linearized ADMM has received particularly wide attention in many areas because of its efficiency and easy implementation. To theoretically guarantee convergence of the linearized ADMM, the step size for the linearized subproblems, or the reciprocal of the linearization parameter, should be sufficiently small. On the other hand, small step sizes decelerate the convergence numerically. Hence, it is interesting to probe the optimal (largest) value of the step size that guarantees convergence of the linearized ADMM. This analysis is lacked in the literature. In this paper, we provide a rigorous mathematical analysis for finding this optimal step size of the linearized ADMM and accordingly set up the optimal version of the linearized ADMM in the convex programming context. The global convergence and worst-case convergence rate measured by the iteration complexity of the optimal version of linearized ADMM are proved as well.

Keywords: Convex programming; Alternating direction method of multipliers; Linearized; Optimal step size; Proximal regularization; Convergence rate (search for similar items in EconPapers)
Date: 2020
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Citations: View citations in EconPapers (5)

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DOI: 10.1007/s10589-019-00152-3

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