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- Title
Quantitative Convergence Analysis of Iterated Expansive, Set-Valued Mappings.
- Authors
Russell Luke, D.; Thao, Nguyen H.; Tam, Matthew K.
- 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.
- Subjects
ITERATIVE methods (Mathematics); FIXED point theory; MATHEMATICAL mappings; SET theory; STOCHASTIC convergence
- Publication
Mathematics of Operations Research, 2018, Vol 43, Issue 4, p1143
- ISSN
0364-765X
- Publication type
Article
- DOI
10.1287/moor.2017.0898