Set regularities and feasibility problems

Abstract

We synthesize and unify notions of regularity, both of individual sets and of collections of sets, as they appear in the convergence theory of projection methods for consistent feasibility problems. Several new characterizations of regularities are presented which shed light on the relations between seemingly different ideas and point to possible necessary conditions for local linear convergence of fundamental algorithms.

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Notes

  1. 1.

    We are particulary indebted to Alex Ioffe for thoughtful and persuasive discussions.

  2. 2.

    We refer on several occasions to the preprint [22] because some definitions and results present there and used in the current article are not included in the published version [23].

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Author information

Affiliations

Authors

Corresponding author

Correspondence to D. Russell Luke.

Additional information

AYK was supported by Australian Research Council, project DP160100854.

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

NHT was supported by German Israeli Foundation Grant G-1253-304.6.

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Kruger, A.Y., Luke, D.R. & Thao, N.H. Set regularities and feasibility problems. Math. Program. 168, 279–311 (2018). https://doi.org/10.1007/s10107-016-1039-x

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Keywords

  • Alternating projections
  • CHIP
  • Clarke regularity
  • Douglas–Rachford
  • Hölder regularity
  • Metric regularity
  • Normal cone
  • Normal qualification condition
  • Prox-regularity
  • Transversality
  • Weak-sharp minima

Mathematics Subject Classification

  • Primary 49J53
  • 65K10
  • Secondary 49K40
  • 49M05
  • 49M37
  • 65K05
  • 90C30