Necessary conditions for linear convergence of iterated expansive, set-valued mappings

Journal article (2018)

Authors

D. Russell Luke University of Göttingen
Marc Teboulle Tel Aviv University
Hieu Thao Nguyen Cantho University, Team Raf Van de Plas - Mechanical, Maritime and Materials Engineering
Research Group
Team Raf Van de Plas (Mechanical, Maritime and Materials Engineering) (TU Delft)
Copyright: © 2018 D. Russell Luke, Marc Teboulle, Hieu Thao Nguyen
DOI: https://doi.org/10.1007/s10107-018-1343-8
Feasibility Transversality Subtransversality Almost averaged mappings Averaged operators Calmness Cyclic projections Elemental regularity Fejér monotone Fixed point iteration Fixed points Metric regularity Metric subregularity Nonconvex Nonexpansive
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Published Date

2018

Language

English

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Faculty

Mechanical, Maritime and Materials Engineering

Department

Delft Center for Systems and Control

Research Group

Team Raf Van de Plas

Abstract

We present necessary conditions for monotonicity of fixed point iterations of mappings that may violate the usual nonexpansive property. Notions of linear-type monotonicity of fixed point sequences—weaker than Fejér monotonicity—are shown to imply metric subregularity. This, together with the almost averaging property recently introduced by Luke et al. (Math Oper Res, 2018. https://doi.org/10.1287/moor.2017.0898), guarantees linear convergence of the sequence to a fixed point. We specialize these results to the alternating projections iteration where the metric subregularity property takes on a distinct geometric characterization of sets at points of intersection called subtransversality. Subtransversality is shown to be necessary for linear convergence of alternating projections for consistent feasibility.