Quantitative Convergence Analysis of Iterated Expansive, Set-Valued Mappings

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​Quantitative Convergence Analysis of Iterated Expansive, Set-Valued Mappings​
Thao, N. H. ; Tam, M. K.   & Luke, R. ​ (2018) 
Mathematics of Operations Research43(4) pp. 1143​-1176​.​

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Thao, Nguyen H. 
Tam, Matthew K. 
Luke, Russell 
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.
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Mathematics of Operations Research 
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