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.
- Alternating direction method of multipliers
- Convex optimization
- Global linear convergence