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Conical averagedness and convergence analysis of fixed point algorithms

Author

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  • Sedi Bartz

    (University of Massachusetts Lowell)

  • Minh N. Dao

    (Federation University Australia)

  • Hung M. Phan

    (University of Massachusetts Lowell)

Abstract

We study a conical extension of averaged nonexpansive operators and the role it plays in convergence analysis of fixed point algorithms. Various properties of conically averaged operators are systematically investigated, in particular, the stability under relaxations, convex combinations and compositions. We derive conical averagedness properties of resolvents of generalized monotone operators. These properties are then utilized in order to analyze the convergence of the proximal point algorithm, the forward–backward algorithm, and the adaptive Douglas–Rachford algorithm. Our study unifies, improves and casts new light on recent studies of these topics.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jglopt:v:82:y:2022:i:2:d:10.1007_s10898-021-01057-4
    DOI: 10.1007/s10898-021-01057-4
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    References listed on IDEAS

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    1. Regina S. Burachik & Alfredo N. Iusem, 2008. "Set-Valued Mappings and Enlargements of Monotone Operators," Springer Optimization and Its Applications, Springer, number 978-0-387-69757-4, June.
    2. 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.
    3. Jean-Philippe Vial, 1983. "Strong and Weak Convexity of Sets and Functions," Mathematics of Operations Research, INFORMS, vol. 8(2), pages 231-259, May.
    4. Regina S. Burachik & Alfredo N. Iusem, 2008. "Enlargements of Monotone Operators," Springer Optimization and Its Applications, in: Set-Valued Mappings and Enlargements of Monotone Operators, chapter 0, pages 161-220, Springer.
    5. VIAL, Jean-Philippe, 1983. "Strong and weak convexity of sets and functions," LIDAM Reprints CORE 529, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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    Cited by:

    1. Chee-Khian Sim, 2023. "Convergence Rates for the Relaxed Peaceman-Rachford Splitting Method on a Monotone Inclusion Problem," Journal of Optimization Theory and Applications, Springer, vol. 196(1), pages 298-323, January.
    2. Sedi Bartz & Rubén Campoy & Hung M. Phan, 2022. "An Adaptive Alternating Direction Method of Multipliers," Journal of Optimization Theory and Applications, Springer, vol. 195(3), pages 1019-1055, December.
    3. Lien T. Nguyen & Andrew Eberhard & Xinghuo Yu & Alexander Y. Kruger & Chaojie Li, 2024. "Finite-Time Nonconvex Optimization Using Time-Varying Dynamical Systems," Journal of Optimization Theory and Applications, Springer, vol. 203(1), pages 844-879, October.

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