On the Douglas-Rachford Algorithm for Solving Possibly Inconsistent Optimization Problems

• 成果类型: Article
• 署名作者: Bauschke, Heinz H.; Moursi, Walaa M.
• 署名单位: University of British Columbia; University of Waterloo; Egyptian Knowledge Bank (EKB); Mansoura University
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• DOI: 10.1287/moor.2022.1347
• 发表日期: 2024
• 关键词: sum; Operators; BEHAVIOR

摘要

More than 40 years ago, Lions and Mercier introduced in a seminal paper the Douglas-Rachford algorithm. Today, this method is well-recognized as a classic and highly successful splitting method to find minimizers of the sum of two (not necessarily smooth) convex functions. Whereas the underlying theory has matured, one case remains a mystery: the behavior of the shadow sequence when the given functions have disjoint domains. Building on previous work, we establish for the first time weak and value convergence of the shadow sequence generated by the Douglas-Rachford algorithm in a setting of unprecedented generality. The weak limit point is shown to solve the associated normal problem, which is a minimal perturbation of the original optimization problem. We also present new results on the geometry of the minimal displacement vector.