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.
Original language | English |
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Pages (from-to) | 1143-1176 |
Journal | Mathematics of Operations Research |
Volume | 43 |
Issue number | 4 |
DOIs | |
Publication status | Published - 2018 |
Keywords
- Analysis of algorithms
- Feasibility
- Fixed points
- Kurdyka-Lojasiewicz inequality
- Linear convergence
- Metric regularity
- Nonconvex
- Nonsmooth
- Proximal algorithms
- Subtransversality
- Transversality