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

Yair Censor (yairmath.haifa.ac.il)
Tommy Elfving (toelfmai.liu.se)
Nirit Kopf (nirit.kopfintel.com)
Thomas Bortfeld (TBORTFELDPARTNERS.ORG)

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 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.

Keywords: Multiple-sets split feasibility; proximity function; Bregman projections; inverse problems; intensity-modulated radiation therapy.

Category 1: Convex and Nonsmooth Optimization (Convex Optimization )

Category 2: Applications -- Science and Engineering

Category 3: Applications -- Science and Engineering (Biomedical Applications )

Citation: Inverse Problems, Vol. 21 (2005), pp. 2071-2084.

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Entry Submitted: 11/22/2005
Entry Accepted: 11/25/2005
Entry Last Modified: 11/25/2005

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