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

Abstract

The multiple-sets split feasibility problem requires to find 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 * Accepted for publication in the journal Inverse Problems. 1 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 (IMRT) treatment planning is studied in a separate companion paper and is here only briefly described.

Strong and weak convergence theorems for split equilibrium problems and fixed point problems in Banach spaces

“…For more results on split feasibility problems, the readers refer to [3,4,6,9,29,34,35]. Now, by Theorem 3.1 and Theorem 4.2, we give the following results on split feasibility problems in Banach spaces: Theorem 5.1.…”

“…Recently, split feasibility problems [3,4,6,9,29,34,35], split variational inequality problems [10,21] and split equilibrium problems [2,17,31] have been investigated by many authors. However, most of the results on these kinds of these problems are investigated only in Hilbert spaces, only a few works are considered in Banach spaces.…”

Iterative algorithms for finding minimum-norm fixed point of a finite family of nonexpansive mappings and applications

“…Recall that the multiple-set split feasibility problem (MSSFP, for short), which was first introduced by Censor et al [5]: 1) where N, M ≥ 1 are integers,…”

“…To solve the MSSFP (4.1), Censor et al [5] first proposed a gradient projection algorithm. However, this iterative algorithm could not reduce to the CQ iterative algorithm [3] for solving the SFP (4.2).…”

Convergence analysis of a Halpern-like iterative algorithm in Hilbert spaces

“…It has been found that the split feasibility problem can also be used in various disciplines such as image restoration, computer tomograph and radiation therapy treatment planning [4,5,7].…”