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

Author

Listed:
  • Minh N. Dao

    (University of Newcastle)

  • Matthew K. Tam

    (Universität Göttingen)

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.

Suggested Citation

  • Minh N. Dao & Matthew K. Tam, 2019. "Union Averaged Operators with Applications to Proximal Algorithms for Min-Convex Functions," Journal of Optimization Theory and Applications, Springer, vol. 181(1), pages 61-94, April.
    • Handle: RePEc:spr:joptap:v:181:y:2019:i:1:d:10.1007_s10957-018-1443-x
    DOI: 10.1007/s10957-018-1443-x
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    References listed on IDEAS

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    1. Minh N. Dao & Hung M. Phan, 2018. "Linear convergence of the generalized Douglas–Rachford algorithm for feasibility problems," Journal of Global Optimization, Springer, vol. 72(3), pages 443-474, November.
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    Cited by:

    1. Sedi Bartz & Minh N. Dao & Hung M. Phan, 2022. "Conical averagedness and convergence analysis of fixed point algorithms," Journal of Global Optimization, Springer, vol. 82(2), pages 351-373, February.
    2. Alexander J. Zaslavski, 2023. "Global Convergence of Algorithms Based on Unions of Non-Expansive Maps," Mathematics, MDPI, vol. 11(14), pages 1-11, July.

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