Solving Composite Fixed Point Problems with Block Updates

TL;DR

This paper introduces a new block update method for composite fixed point problems that allows convergence without activating all operators at each step, applicable to various nonlinear and nonsmooth problems.

Contribution

It proposes a novel block update algorithm for composite fixed point problems that maintains convergence, extending current methods that require activating all operators.

Findings

Achieves weak, strong, and linear convergence results.

Applicable to monotone inclusions and inconsistent feasibility problems.

Demonstrates effectiveness in data science minimization tasks.

Abstract

Various strategies are available to construct iteratively a common fixed point of nonexpansive operators by activating only a block of operators at each iteration. In the more challenging class of composite fixed point problems involving operators that do not share common fixed points, current methods require the activation of all the operators at each iteration, and the question of maintaining convergence while updating only blocks of operators is open. We propose a method that achieves this goal and analyze its asymptotic behavior. Weak, strong, and linear convergence results are established by exploiting a connection with the theory of concentrating arrays. Applications to several nonlinear and nonsmooth analysis problems are presented, ranging from monotone inclusions and inconsistent feasibility problems, to minimization problems arising in data science.