@article{4e8787a9c1e04e24b818716168379a50,
title = "Necessary conditions for linear convergence of iterated expansive, set-valued mappings",
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 Fej{\'e}r 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.",
keywords = "Almost averaged mappings, Averaged operators, Calmness, Cyclic projections, Elemental regularity, Feasibility, Fej{\'e}r monotone, Fixed point iteration, Fixed points, Metric regularity, Metric subregularity, Nonconvex, Nonexpansive, Subtransversality, Transversality",
author = "Luke, {D. Russell} and Marc Teboulle and Thao, {Nguyen H.}",
note = "Publisher Copyright: {\textcopyright} 2018, Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society.",
year = "2020",
month = mar,
day = "1",
doi = "10.1007/s10107-018-1343-8",
language = "אנגלית",
volume = "180",
pages = "1--31",
journal = "Mathematical Programming",
issn = "0025-5610",
publisher = "Springer-Verlag GmbH and Co. KG",
number = "1-2",
}