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

D. Russell Luke*, Marc Teboulle, Nguyen H. Thao

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

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.

Original languageEnglish
Pages (from-to)1-31
Number of pages31
JournalMathematical Programming
Volume180
Issue number1-2
DOIs
StatePublished - 1 Mar 2020

Funding

FundersFunder number
Deutsche ForschungsgemeinschaftGRK 2088 TP-B5
German-Israeli Foundation for Scientific Research and DevelopmentG-1253-304.6/2014
Israel Science Foundation1844-16

    Keywords

    • Almost averaged mappings
    • Averaged operators
    • Calmness
    • Cyclic projections
    • Elemental regularity
    • Feasibility
    • Fejér monotone
    • Fixed point iteration
    • Fixed points
    • Metric regularity
    • Metric subregularity
    • Nonconvex
    • Nonexpansive
    • Subtransversality
    • Transversality

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