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A generalized block-iterative projection method for the common fixed point problem induced by cutters

Author

Listed:
  • Yair Censor

    (University of Haifa)

  • Daniel Reem

    (University of Haifa)

  • Maroun Zaknoon

    (The Arab Academic College for Education)

Abstract

The block-iterative projections (BIP) method of Aharoni and Censor [Block-iterative projection methods for parallel computation of solutions to convex feasibility problems, Linear Algebra and its Applications 120, (1989), 165–175] is an iterative process for finding asymptotically a point in the nonempty intersection of a family of closed convex subsets. It employs orthogonal projections onto the individual subsets in an algorithmic regime that uses “blocks” of operators and has great flexibility in constructing specific algorithms from it. We extend this algorithmic scheme to handle a family of continuous cutter operators and to find a common fixed point of them. Since the family of continuous cutters includes several important specific operators, our generalized scheme, which ensures global convergence and retains the flexibility of BIP, can handle, in particular, metric (orthogonal) projectors and continuous subgradient projections, which are very important in applications. We also allow a certain kind of adaptive perturbations to be included, and along the way we derive a perturbed Fejér monotonicity lemma which is of independent interest.

Suggested Citation

  • Yair Censor & Daniel Reem & Maroun Zaknoon, 2022. "A generalized block-iterative projection method for the common fixed point problem induced by cutters," Journal of Global Optimization, Springer, vol. 84(4), pages 967-987, December.
    • Handle: RePEc:spr:jglopt:v:84:y:2022:i:4:d:10.1007_s10898-022-01175-7
    DOI: 10.1007/s10898-022-01175-7
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    References listed on IDEAS

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    1. Alexander J. Zaslavski, 2016. "Approximate Solutions of Common Fixed-Point Problems," Springer Optimization and Its Applications, Springer, number 978-3-319-33255-0, September.
    2. Heinz H. Bauschke & Caifang Wang & Xianfu Wang & Jia Xu, 2015. "On the Finite Convergence of a Projected Cutter Method," Journal of Optimization Theory and Applications, Springer, vol. 165(3), pages 901-916, June.
    3. Aviv Gibali & Karl-Heinz Küfer & Daniel Reem & Philipp Süss, 2018. "A generalized projection-based scheme for solving convex constrained optimization problems," Computational Optimization and Applications, Springer, vol. 70(3), pages 737-762, July.
    4. Andrzej Cegielski & Yair Censor, 2011. "Opial-Type Theorems and the Common Fixed Point Problem," Springer Optimization and Its Applications, in: Heinz H. Bauschke & Regina S. Burachik & Patrick L. Combettes & Veit Elser & D. Russell Luke & Henry (ed.), Fixed-Point Algorithms for Inverse Problems in Science and Engineering, chapter 0, pages 155-183, Springer.
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