The multiple-sets split feasibility problem and its applications for inverse problems

TLDR

The multiple‑sets split feasibility problem seeks a point nearest to a family of convex sets in one space whose image under a linear map is nearest to another family of convex sets, generalizing both convex feasibility and two‑sets split feasibility, and modeling many inverse problems with domain and range constraints. The authors propose a projection algorithm that minimizes a proximity function measuring distance to all sets, and extend it to Bregman distance‑based proximity functions. The algorithm generalizes previous split feasibility methods by incorporating Bregman distances within a projection framework. The method is applied to intensity‑modulated radiation therapy treatment planning, with results presented in a companion paper.

Abstract

The multiple-sets split feasibility problem requires finding a point closest to a family of closed convex sets in one space such that its image under a linear transformation will be closest to another family of closed convex sets in the image space. It can be a model for many inverse problems where constraints are imposed on the solutions in the domain of a linear operator as well as in the operator's range. It generalizes the convex feasibility problem as well as the two-sets split feasibility problem. We propose a projection algorithm that minimizes a proximity function that measures the distance of a point from all sets. The formulation, as well as the algorithm, generalize earlier work on the split feasibility problem. We offer also a generalization to proximity functions with Bregman distances. Application of the method to the inverse problem of intensity-modulated radiation therapy treatment planning is studied in a separate companion paper and is here only described briefly.