Quantitative Convergence Analysis of Iterated Expansive, Set-Valued Mappings
2018 | journal article. A publication with affiliation to the University of Göttingen.
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Thao, Nguyen H., Matthew K. Tam, and Russell Luke. "Quantitative Convergence Analysis of Iterated Expansive, Set-Valued Mappings." Mathematics of Operations Research 43, no. 4 (2018): 1143-1176. https://doi.org/10.1287/moor.2017.0898.
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- Authors
- Thao, Nguyen H. ; Tam, Matthew K. ; Luke, Russell
- Abstract
- We develop a framework for quantitative convergence analysis of Picard iterations of expansive set-valued fixed point mappings. There are two key components of the analysis. The first is a natural generalization of single-valued averaged mappings to expansive set-valued mappings that characterizes a type of strong calmness of the fixed point mapping. The second component to this analysis is an extension of the well-established notion of metric subregularity—or inverse calmness—of the mapping at fixed points. Convergence of expansive fixed point iterations is proved using these two properties, and quantitative estimates are a natural by-product of the framework. To demonstrate the application of the theory, we prove, for the first time, a number of results showing local linear convergence of nonconvex cyclic projections for inconsistent (and consistent) feasibility problems, local linear convergence of the forward-backward algorithm for structured optimization without convexity, strong or otherwise, and local linear convergence of the Douglas-Rachford algorithm for structured nonconvex minimization. This theory includes earlier approaches for known results, convex and nonconvex, as special cases.
- Issue Date
- 2018
- Journal
- Mathematics of Operations Research
- Project
- RTG 2088: Research Training Group 2088 Discovering structure in complex data: Statistics meets Optimization and Inverse Problems
- Organization
- Institut für Numerische und Angewandte Mathematik
- Working Group
- RG Luke (Continuous Optimization, Variational Analysis and Inverse Problems)
- Language
- English