On the global linear convergence of the ADMM with multiblock variables

The alternating direction method of multipliers (ADMM) has been widely used for solving structured convex optimization problems. In particular, the ADMM can solve convex programs that minimize the sum of N convex functions whose variables are linked by some linear constraints. While the convergence of the ADMM for N = 2 was well established in the literature, it remained an open problem for a long time whether the ADMM for N ≥ 3 is still convergent. Recently, it was shown in [Chen et al., Math. Program. (2014), DOI 10.1007/s10107-014-0826-5.] that without additional conditions the ADMM for N ≥ 3 generally fails to converge. In this paper, we show that under some easily verifiable and reasonable conditions the global linear convergence of the ADMM when N ≥ 3 can still be ensured, which is important since the ADMM is a popular method for solving large-scale multiblock optimization models and is known to perform very well in practice even when N ≥ 3. Our study aims to offer an explanation for this phenomenon.