No 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.

We propose and study a unified model for handling dose constraints (physical dose, equivalent uniform dose (EUD), etc) and radiation source constraints in a single mathematical framework based on the split feasibility problem. The model does not impose on the constraints an exogenous objective (merit) function. The optimization algorithm minimizes a weighted proximity function that measures the sum of the squares of the distances to the constraint sets. This guarantees convergence to a feasible solution point if the split feasibility problem is consistent (i.e., has a solution), or, otherwise, convergence to a solution that minimally violates the physical dose constraints and EUD constraints. We present computational results that demonstrate the validity of the model and the power of the proposed algorithmic scheme.