Journal
SIAM JOURNAL ON OPTIMIZATION
Volume 22, Issue 2, Pages 313-340Publisher
SIAM PUBLICATIONS
DOI: 10.1137/110822347
Keywords
alternating direction method; convex programming; Gaussian back substitution; separable structure
Categories
Funding
- NSFC [10971095, 91130007]
- Cultivation Fund of KSTIP-MOEC [708044]
- MOEC fund [20110091110004]
- General Research Fund of Hong Kong [HKBU203311]
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We consider the linearly constrained separable convex minimization problem whose objective function is separable into m individual convex functions with nonoverlapping variables. A Douglas Rachford alternating direction method of multipliers (ADM) has been well studied in the literature for the special case of m = 2. But the convergence of extending ADM to the general case of m >= 3 is still open. In this paper, we show that the straightforward extension of ADM is valid for the general case of m >= 3 if it is combined with a Gaussian back substitution procedure. The resulting ADM with Gaussian back substitution is a novel approach towards the extension of ADM from m = 2 to m >= 3, and its algorithmic framework is new in the literature. For the ADM with Gaussian back substitution, we prove its convergence via the analytic framework of contractive-type methods, and we show its numerical efficiency by some application problems.
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