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Union Averaged Operators with Applications to Proximal Algorithms for Min-Convex Functions

  • Minh N. Dao
  • Matthew K. TamEmail author
Article
  • 219 Downloads

Abstract

In this paper, we introduce and study a class of structured set-valued operators, which we call union averaged nonexpansive. At each point in their domain, the value of such an operator can be expressed as a finite union of single-valued averaged nonexpansive operators. We investigate various structural properties of the class and show, in particular, that is closed under taking unions, convex combinations, and compositions, and that their fixed point iterations are locally convergent around strong fixed points. We then systematically apply our results to analyze proximal algorithms in situations, where union averaged nonexpansive operators naturally arise. In particular, we consider the problem of minimizing the sum two functions, where the first is convex and the second can be expressed as the minimum of finitely many convex functions.

Keywords

Admissible control Averaged operator Fixed point iteration Local convergence Proximal algorithms Set-valued map 

Mathematics Subject Classification

90C26 7H10 47H04 

Notes

Acknowledgements

The authors are thankful to the anonymous referee for their constructive comments and suggestions. MND was partially supported by the Australian Research Council (ARC) Discovery Project DP160101537 and by the Priority Research Centre for Computer-Assisted Research Mathematics and its Applications (CARMA) at the University of Newcastle. He wishes to acknowledge the hospitality and the support of D. Russell Luke during his visit to the Universität Göttingen. MKT was partially supported by a Postdoctoral Fellowship from the Alexander von Humboldt Foundation.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.CARMAUniversity of NewcastleCallaghanAustralia
  2. 2.Institut für Numerische und Angewandte MathematikUniversität GöttingenGöttingenGermany

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