Generalized monotone operators and their averaged resolvents

• 成果类型: Article
• 署名作者: Bauschke, Heinz H.; Moursi, Walaa M.; Wang, Xianfu
• 署名单位: University of British Columbia; University of Waterloo
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-020-01500-6
• 发表日期: 2021
• 页码: 55-74
• 关键词: proximal point algorithm; subdifferentials; SUM

摘要

The correspondence between the monotonicity of a (possibly) set-valued operator and the firm nonexpansiveness of its resolvent is a key ingredient in the convergence analysis of many optimization algorithms. Firmly nonexpansive operators form a proper subclass of the more general-but still pleasant from an algorithmic perspective-class of averaged operators. In this paper, we introduce the new notion of conically nonexpansive operators which generalize nonexpansive mappings. We characterize averaged operators as being resolvents of comonotone operators under appropriate scaling. As a consequence, we characterize the proximal point mappings associated with hypoconvex functions as cocoercive operators, or equivalently; as displacement mappings of conically nonexpansive operators. Several examples illustrate our analysis and demonstrate tightness of our results.