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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$$(N\\ge 3)$$(Nź3) 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 $$O(1/\\epsilon ^2)$$O(1/∈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 $$O(1/\\epsilon )$$O(1/∈) iterations when it is applied to solve a certain sharing problem, under the condition that the augmented Lagrangian function satisfies the Kurdyka---źojasiewicz 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.

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