Scaled relative graphs: nonexpansive operators via 2D Euclidean geometry

• 成果类型: Article
• 署名作者: Ryu, Ernest K.; Hannah, Robert; Yin, Wotao
• 署名单位: Seoul National University (SNU); University of California System; University of California Los Angeles
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-021-01639-w
• 发表日期: 2022
• 页码: 569-619
• 关键词: backward splitting method; monotone inclusions; rachford; CONVERGENCE; performance; REGULARITY; algorithm; SUM

摘要

Many iterative methods in applied mathematics can be thought of as fixed-point iterations, and such algorithms are usually analyzed analytically, with inequalities. In this paper, we present a geometric approach to analyzing contractive and non-expansive fixed point iterations with a new tool called the scaled relative graph. The SRG provides a correspondence between nonlinear operators and subsets of the 2D plane. Under this framework, a geometric argument in the 2D plane becomes a rigorous proof of convergence.