Abstract Version of CARP Algorithm

In this chapter we study the convergence of an abstract version of the algorithm which is called in the literature as component-averaged row projections or CARP. This algorithm was introduced for solving a convex feasibility problem in a finite-dimensional space, when a given collection of sets is divided into blocks in such a manner that all sets belonging to every block are subsets of a vector subspace associated with the block. All the blocks are processed in parallel and the algorithm operates in vector subspaces of the whole vector space. This method becomes efficient, in particular, when the dimensions of the subspaces are essentially smaller than the dimension of the whole space. In the chapter we study CARP for problems in a normed space, which is not necessarily finite-dimensional. Our main goal is to obtain an approximate solution of the problem using perturbed algorithms. We show that the inexact dynamic string-averaging algorithm generates an approximate solution if perturbations are summable. We also show that if the mappings are nonexpansive and the perturbations are sufficiently small, then the inexact dynamic string-averaging algorithm produces approximate solutions.