Necessary conditions for linear convergence of iterated expansive, set-valued mappings

• 成果类型: Article
• 署名作者: Luke, D. Russell; Teboulle, Marc; Thao, Nguyen H.
• 署名单位: University of Gottingen; Tel Aviv University; Delft University of Technology; Can Tho University
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-018-1343-8
• 发表日期: 2020
• 页码: 1-31
• 关键词: alternating projections; metric regularity; collections; algorithms

摘要

We present necessary conditions for monotonicity of fixed point iterations of mappings that may violate the usual nonexpansive property. Notions of linear-type monotonicity of fixed point sequences-weaker than Fejer monotonicity-are shown to imply metric subregularity. This, together with the almost averaging property recently introduced by Luke et al. (Math Oper Res, 2018. 10.1287/moor.2017.0898), guarantees linear convergence of the sequence to a fixed point. We specialize these results to the alternating projections iteration where the metric subregularity property takes on a distinct geometric characterization of sets at points of intersection called subtransversality. Subtransversality is shown to be necessary for linear convergence of alternating projections for consistent feasibility.