Monotone operator theory in convex optimization

• 成果类型: Article; Proceedings Paper
• 署名作者: Combettes, Patrick L.
• 署名单位: North Carolina State University
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-018-1303-3
• 发表日期: 2018
• 页码: 177-206
• 关键词: proximal point algorithm; least-squares solutions; splitting method; analyse fonctionnelle; thresholding algorithm; nonlinear operators; partial inverses; signal recovery; DECOMPOSITION; CONVERGENCE

摘要

Several aspects of the interplay between monotone operator theory and convex optimization are presented. The crucial role played by monotone operators in the analysis and the numerical solution of convex minimization problems is emphasized. We review the properties of subdifferentials as maximally monotone operators and, in tandem, investigate those of proximity operators as resolvents. In particular, we study new transformations which map proximity operators to proximity operators, and establish connections with self-dual classes of firmly nonexpansive operators. In addition, new insights and developments are proposed on the algorithmic front.