  


On the Sublinear Convergence Rate of MultiBlock ADMM
Tianyi Lin (lintyse.cuhk.edu.hk) Abstract: The alternating direction method of multipliers (ADMM) is widely used in solving structured convex optimization problems. Despite of its success in practice, the convergence of the standard ADMM for minimizing the sum of $N$ $(N\geq 3)$ convex functions whose variables are linked by linear constraints, has remained unclear for a very long time. Recently, Chen et. al provided a counterexample showing that the ADMM for $N\geq 3$ may fail to converge without further conditions. Since the ADMM for $N\geq 3$ has been very successful when applied to many problems arising from real practice, it is worth further investigating under what kind of sufficient conditions it can be guaranteed to converge. In this paper, we present such sufficient conditions that can guarantee the sublinear convergence rate for the ADMM for $N\geq 3$. Specifically, we show that if one of the functions is convex (not necessarily strongly convex) and the other $N1$ functions are strongly convex, and the penalty parameter lies in a certain region, the ADMM converges with rate $O(1/t)$ in a certain ergodic sense, and $o(1/t)$ in a certain nonergodic sense, where $t$ denotes the number of iterations. As a byproduct, we also provide a simple proof for the $O(1/t)$ convergence rate of twoblock ADMM in terms of both objective error and constraint violation, without assuming any condition on the penalty parameter and strong convexity on the functions. Keywords: Alternating Direction Method of Multipliers, Convergence Rate, Convex Optimization Category 1: Convex and Nonsmooth Optimization Citation: Download: [PDF] Entry Submitted: 08/19/2014 Modify/Update this entry  
Visitors  Authors  More about us  Links  
Subscribe, Unsubscribe Digest Archive Search, Browse the Repository

Submit Update Policies 
Coordinator's Board Classification Scheme Credits Give us feedback 
Optimization Journals, Sites, Societies  