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.
Bibliographical noteFunding Information:
The authors thank the referees for their comments, which helped improve the paper. The work of the first and second authors was supported by the ISF-NSFC joint research plan, Grant Number 2874/19.
© 2022, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.
- Block iterative projections (BIP)
- Common fixed point problem
- Weight function
ASJC Scopus subject areas
- Computer Science Applications
- Control and Optimization
- Management Science and Operations Research
- Applied Mathematics