On the Range of the Douglas-Rachford Operator

• 成果类型: Article
• 署名作者: Bauschke, Heinz H.; Hare, Warren L.; Moursi, Walaa M.
• 署名单位: University of British Columbia; Egyptian Knowledge Bank (EKB); Mansoura University
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• DOI: 10.1287/moor.2015.0759
• 发表日期: 2016
• 页码: 884-897
• 关键词: monotone-operators; SUM; Duality; paramonotonicity

摘要

The problem of finding a minimizer of the sum of two convex functions-or, more generally, that of finding a zero of the sum of two maximally monotone operators-is of central importance in variational analysis. Perhaps the most popular method of solving this problem is the Douglas-Rachford splitting method. Surprisingly, little is known about the range of the Douglas-Rachford operator. In this paper, we set out to study this range systematically. We prove that for 3* monotone operators a very pleasing formula can be found that reveals the range to be nearly equal to a simple set involving the domains and ranges of the underlying operators. A similar formula holds for the range of the corresponding displacement mapping. We discuss applications to subdifferential operators, to the infimal displacement vector, and to firmly nonexpansive mappings. Various examples and counterexamples are presented, including some concerning the celebrated Brezis-Haraux theorem.