ON THE BEHAVIOR OF THE DOUGLAS RACHFORD ALGORITHM FOR MINIMIZING A CONVEX FUNCTION SUBJECT TO A LINEAR CONSTRAINT

作     者：Bauschke, Heinz H. Moursi, Walaa M. 

作者机构：Univ British Columbia Math Kelowna BC V1V 1V7 Canada Univ Waterloo Dept Combinator & Optimizat Waterloo ON N2L 3G1 Canada 

出 版 物：《SIAM JOURNAL ON OPTIMIZATION》 (工业与应用数学会最优化杂志)

年 卷 期：2020年第30卷第3期

页      面：2559-2576页

核心收录：

学科分类：07[理学] 070104[理学-应用数学] 0701[理学-数学] 

基　　金：Natural Sciences and Engineering Research Council of Canada 

主　　题：convex optimization problem Douglas-Rachford splitting inconsistent constrained optimization least squares solution normal problem parallel splitting method projection operator proximal mapping 

摘      要：The Douglas Rachford algorithm (DRA) is a powerful optimization method for minimizing the sum of two convex (not necessarily smooth) functions. The vast majority of previous research dealt with the case when the sum has at least one minimizer. In the absence of minimizers, it was recently shown that for the case of two indicator functions, the DRA converges to a best approximation solution. In this paper, we present a new convergence result on the DRA applied to the problem of minimizing a convex function subject to a linear constraint. Indeed, a normal solution may be found even when the domain of the objective function and the linear subspace constraint have no point in common. As an important application, a new parallel splitting result is provided. We also illustrate our results through various examples.