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arXiv · 1605.05725

Quantitative convergence analysis of iterated expansive, set-valued mappings

D. Russell Luke·Nguyen H. Thao·Matthew K. Tam
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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 byproduct 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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D. Russell Luke, Nguyen H. Thao, Matthew K. Tam. 2017-03-25. Quantitative convergence analysis of iterated expansive, set-valued mappings. https://doi.org/10.1287/moor.2017.0898

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