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Conical averagedness and convergence analysis of fixed point algorithms

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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.

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Acknowledgements

We would like to thank two anonymous referees for their careful review and helpful comments. SB was partially supported by a UMass Lowell faculty startup grant. MND was partially supported by Discovery Projects 160101537 and 190100555 from the Australian Research Council. HMP was partially supported by Autodesk, Inc. via a gift made to the Department of Mathematical Sciences, UMass Lowell.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Kennedy College of Sciences, University of Massachusetts Lowell, Lowell, MA, 01854, USA

    Sedi Bartz & Hung M. Phan

  2. School of Engineering, Information Technology and Physical Sciences, Federation University Australia, Ballarat, VIC, 3353, Australia

    Minh N. Dao

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  1. Sedi Bartz
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  2. Minh N. Dao
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  3. Hung M. Phan
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Correspondence to Minh N. Dao.

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Bartz, S., Dao, M.N. & Phan, H.M. Conical averagedness and convergence analysis of fixed point algorithms. J Glob Optim 82, 351–373 (2022). https://doi.org/10.1007/s10898-021-01057-4

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