String-Averaging Algorithms for Convex Feasibility with Infinitely Many Sets

TL;DR

This paper extends string-averaging algorithms for convex feasibility problems to cases with infinitely many convex sets, providing convergence guarantees and perturbation resilience for practical applications.

Contribution

It introduces new theorems ensuring convergence of string-averaging algorithms with infinitely many convex sets, a case not thoroughly addressed before.

Findings

Algorithms converge to a point in the intersection of infinitely many convex sets.

The proposed algorithms are perturbation resilient, enabling superiorization.

Theoretical results extend prior finite-set algorithms to infinite collections.

Abstract

Algorithms for convex feasibility find or approximate a point in the intersection of given closed convex sets. Typically there are only finitely many convex sets, but the case of infinitely many convex sets also has some applications. In this context, a \emph{string} is a finite sequence of points each of which is obtained from the previous point by considering one of the convex sets. In a \emph{string-averaging} algorithm, an iterative step from a first point to a second point computes a number of strings, all starting with the first point, and then calculates a (weighted) average of those strings' end-points: This average is that iterative step's second point, which is used as the first point for the next iterative step. For string-averaging algorithms based on strings in which each point either is the projection of the previous point on one of the convex sets or is equal to the previous point, we present theorems that provide answers to the following question: "How can the iterative steps be specified so that the string-averaging algorithm generates a convergent sequence whose limit lies in the intersection of the sets of a given convex feasibility problem?" This paper focuses on the case where the given collection of convex sets is infinite, whereas prior work on the same question that we are aware of has assumed the collection of convex sets is finite (or has been applicable only to a small subset of the algorithms we consider). The string-averaging algorithms that are shown to generate a convergent sequence whose limit lies in the intersection are also shown to be perturbation resilient, so they can be superiorized to generate sequences that also converge to a point of the intersection but in a manner that is superior with respect to some application-dependent criterion.