期刊
SIAM JOURNAL ON OPTIMIZATION
卷 25, 期 3, 页码 1478-1497出版社
SIAM PUBLICATIONS
DOI: 10.1137/140971178
关键词
alternating direction method of multipliers; global linear convergence; convex optimization
资金
- Hong Kong Research Grants Council General Research Fund Early Career Scheme [CUHK 439513]
- NSF grant [CMMI-1161242]
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.
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