- Iteration Complexity Analysis of Multi-Block ADMM for a Family of Convex Minimization without Strong Convexity Tianyi Lin (tylinse.cuhk.edu.hk) Shiqian Ma (sqmase.cuhk.edu.hk) Shuzhong Zhang (zhangsumn.edu) Abstract: The alternating direction method of multipliers (ADMM) is widely used in solving structured convex optimization problems due to its superior practical performance. On the theoretical side however, a counterexample was shown in [7] indicating that the multi-block ADMM for minimizing the sum of $N$ $(N\geq 3)$ convex functions with $N$ block variables linked by linear constraints may diverge. It is therefore of great interest to investigate further sufficient conditions on the input side which can guarantee convergence for the multi-block ADMM. The existing results typically require the strong convexity on parts of the objective. In this paper, we present convergence and convergence rate results for the multi-block ADMM applied to solve certain $N$-block $(N\geq 3)$ convex minimization problems {\it without requiring strong convexity}. Specifically, we prove the following two results: (1) the multi-block ADMM returns an $\epsilon$-optimal solution within $O(1/\epsilon^2)$ iterations by solving an associated perturbation to the original problem; (2) the multi-block ADMM returns an $\epsilon$-optimal solution within $O(1/\epsilon)$ iterations when it is applied to solve a certain {\it sharing problem}, under the condition that the augmented Lagrangian function satisfies the Kurdyka-{\L}ojasiewicz property, which essentially covers most convex optimization models except for some pathological cases. Keywords: Alternating Direction Method of Multipliers, Convergence Rate, Kurdyka-Lojasiewicz property Category 1: Convex and Nonsmooth Optimization Citation: Download: [PDF]Entry Submitted: 04/13/2015Entry Accepted: 04/13/2015Entry Last Modified: 05/19/2015Modify/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 Optimization Online is supported by the Mathematical Optmization Society.