Iteration Complexity Analysis of Multi-block ADMM for a Family of Convex Minimization Without Strong Convexity

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 Chen et al. (Math Program 155(1):57–79, 2016.) indicating that the multi-block ADMM for minimizing the sum of N \((N\ge 3)\) 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 provide two different ways related to multi-block ADMM that can find an \(\epsilon \)-optimal solution and do not require strong convexity of the objective function. 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; this case can be seen as using multi-block ADMM to solve a modified 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–Łojasiewicz property, which essentially covers most convex optimization models except for some pathological cases; this case can be seen as applying multi-block ADMM to solving a special class of problems.