ALTERNATING DIRECTION METHOD WITH GAUSSIAN BACK SUBSTITUTION FOR SEPARABLE CONVEX PROGRAMMING

作者：He, Bingsheng^*; Tao, Min; Yuan, Xiaoming

来源：SIAM Journal on Optimization, 2012, 22(2): 313-340.

DOI：10.1137/110822347

• alternating direction method
• convex programming
• Gaussian back substitution
• separable structure

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.

• 出版日期2012
• 单位南京大学