Department of Mathematics
Alternating direction method with Gaussian back substitution for separable convex programming
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. © 2012 Society for Industrial and Applied Mathematics.
Alternating direction method, Convex programming, Gaussian back substitution, Separable structure
Source Publication Title
SIAM Journal on Optimization
Society for Industrial and Applied Mathematics
Link to Publisher's Edition
He, Bingsheng, Min Tao, and Xiaoming Yuan. "Alternating direction method with Gaussian back substitution for separable convex programming." SIAM Journal on Optimization 22.2 (2012): 313-340.