Iteration Complexity Analysis of Multi-Block ADMM for a Family of Convex Minimization without Strong Convexity

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 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 sharing problem, under the condition that the augmented Lagrangian function satisfies the Kurdyka-Lojasiewicz property, which essentially covers most convex optimization models except for some pathological cases.