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Asynchronous sequential inertial iterations for common fixed points problems with an application to linear systems

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  • Howard Heaton

    (University of California Los Angeles)

  • Yair Censor

    (University of Haifa)

Abstract

The common fixed points problem requires finding a point in the intersection of fixed points sets of a finite collection of operators. Quickly solving problems of this sort is of great practical importance for engineering and scientific tasks (e.g., for computed tomography). Iterative methods for solving these problems often employ a Krasnosel’skiĭ–Mann type iteration. We present an asynchronous sequential inertial (ASI) algorithmic framework in a Hilbert space to solve common fixed points problems with a collection of nonexpansive operators. Our scheme allows use of out-of-date iterates when generating updates, thereby enabling processing nodes to work simultaneously and without synchronization. This method also includes inertial type extrapolation terms to increase the speed of convergence. In particular, we extend the application of the recent “ARock algorithm” (Peng et al. in SIAM J Sci Comput 38:A2851–A2879, 2016) in the context of convex feasibility problems. Convergence of the ASI algorithm is proven with no assumption on the distribution of delays, except that they be uniformly bounded. Discussion is provided along with a computational example showing the performance of the ASI algorithm applied in conjunction with a diagonally relaxed orthogonal projections (DROP) algorithm for estimating solutions to large linear systems.

Suggested Citation

  • Howard Heaton & Yair Censor, 2019. "Asynchronous sequential inertial iterations for common fixed points problems with an application to linear systems," Journal of Global Optimization, Springer, vol. 74(1), pages 95-119, May.
    • Handle: RePEc:spr:jglopt:v:74:y:2019:i:1:d:10.1007_s10898-019-00747-4
    DOI: 10.1007/s10898-019-00747-4
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    References listed on IDEAS

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    1. Yair Censor & Rafiq Mansour, 2018. "Convergence Analysis of Processes with Valiant Projection Operators in Hilbert Space," Journal of Optimization Theory and Applications, Springer, vol. 176(1), pages 35-56, January.
    2. Jonathan Eckstein, 2017. "A Simplified Form of Block-Iterative Operator Splitting and an Asynchronous Algorithm Resembling the Multi-Block Alternating Direction Method of Multipliers," Journal of Optimization Theory and Applications, Springer, vol. 173(1), pages 155-182, April.
    3. Yair Censor & Wei Chen & Patrick Combettes & Ran Davidi & Gabor Herman, 2012. "On the effectiveness of projection methods for convex feasibility problems with linear inequality constraints," Computational Optimization and Applications, Springer, vol. 51(3), pages 1065-1088, April.
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