The old alternating direction method (ADM) has found many new applications recently, and its empirical efficiency has been well illustrated in various fields. However, the estimate of ADM's convergence rate remains a theoretical challenge for a few decades. In this note, we provide a uniform proof to show the O(1/t) convergence rate for both the original ADM and its linearized variant (known as the split inexact Uzawa method in image processing literature). The proof is based on a variational inequality approach which is novel in the literature, and it is very simple.
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View On the O(1/t) convergence rate of alternating direction method