On Douglas-Rachford operators that fail to be proximal mappings

• 成果类型: Article
• 署名作者: Bauschke, Heinz H.; Schaad, Jason; Wang, Xianfu
• 署名单位: University of British Columbia
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-016-1076-5
• 发表日期: 2018
• 页码: 55-61
• 关键词: point algorithm

摘要

The problem of finding a zero of the sum of two maximally monotone operators is of central importance in optimization. One successful method to find such a zero is the Douglas-Rachford algorithm which iterates a firmly nonexpansive operator constructed from the resolvents of the given monotone operators. In the context of finding minimizers of convex functions, the resolvents are actually proximal mappings. Interestingly, as pointed out by Eckstein in 1989, the Douglas-Rachford operator itself may fail to be a proximal mapping. We consider the class of symmetric linear relations that are maximally monotone and prove the striking result that the Douglas-Rachford operator is generically not a proximal mapping.