Algorithms based on unions of nonexpansive maps

Abstract

In this work, we consider a framework for the analysis of iterative algorithms which can be described in terms of a structured set-valued operator. More precisely, at each point in the ambient space, we assume that the value of the operator can be expressed as a finite union of values of single-valued paracontracting operators. Our main result, which shows that the associated fixed point iteration is locally convergent around strong fixed points, generalises a theorem due to Bauschke and Noll (Arch Math 102(6):589–600, 2014).

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Acknowledgements

This work was supported by a Postdoctoral Fellowship from the Alexander von Humboldt Foundation. The author wishes to thank Yura Malitsky for suggesting Corollary 1 and the two anonymous referees whose comments have improved the manuscript.

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Correspondence to Matthew K. Tam.

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Tam, M.K. Algorithms based on unions of nonexpansive maps. Optim Lett 12, 1019–1027 (2018). https://doi.org/10.1007/s11590-018-1249-7

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Keywords

  • Set-valued map
  • Fixed point iteration
  • Paracontracting operator
  • Sparsity