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

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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.

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Authors and Affiliations

  1. Institut für Numerische und Angewandte Mathematik, Universität Göttingen, 37083, Göttingen, Germany

    D. Russell Luke

  2. School of Mathematical Sciences, Tel Aviv University, 69978, Tel Aviv, Israel

    Marc Teboulle

  3. Delft Center for Systems and Control, Delft University of Technology, 2628CD, Delft, The Netherlands

    Nguyen H. Thao

  4. Department of Mathematics, Teacher College, Cantho University, Cantho, Vietnam

    Nguyen H. Thao

Authors
  1. D. Russell Luke
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  2. Marc Teboulle
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  3. Nguyen H. Thao
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Correspondence to D. Russell Luke.

Additional information

D. Russell Luke was supported in part by German Israeli Foundation Grant G-1253-304.6 and Deutsche Forschungsgemeinschaft Research Training Grant 2088 TP-B5.

Marc Teboulle was supported by German Israeli Foundation Grant G-1253-304.6 and Israel Science Foundation ISF Grant 1844-16.

Nguyen H. Thao was supported by German Israeli Foundation Grant G-1253-304.6.

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Luke, D.R., Teboulle, M. & Thao, N.H. Necessary conditions for linear convergence of iterated expansive, set-valued mappings. Math. Program. 180, 1–31 (2020). https://doi.org/10.1007/s10107-018-1343-8

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  • DOI: https://doi.org/10.1007/s10107-018-1343-8

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