A Generalized Block-Iterative Projection Method for the Common Fixed Point Problem Induced by Cutters

TL;DR

This paper introduces a generalized block-iterative projection method for finding common fixed points of continuous cutter operators, extending previous algorithms to broader classes of operators with guaranteed convergence.

Contribution

It extends the BIP method to handle continuous cutters, including metric projections and subgradient projections, with convergence guarantees and adaptive perturbation handling.

Findings

Ensures global convergence of the generalized BIP scheme.

Handles a broad class of operators including metric and subgradient projections.

Incorporates adaptive perturbations with a new Fejér monotonicity lemma.

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 Fej\'er monotonicity lemma which is of independent interest.