  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.
