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The multiple-sets split feasibility problem and its applications for inverse problems. (English) Zbl 1089.65046
The multiple-sets split feasibility problem generalizing the convex feasibility problem as well as the two-sets split feasibility problem is formulated and an algorithm for its solution is proposed. The 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.
The problem can be formulated as follows. Given closed convex sets $$C_i\subseteq\mathbb{R}^N$$, $$i= 1,\dots, t$$ and closed convex sets $$Q_j\subseteq\mathbb{R}^M$$, $$j= 1,\dots, r$$, in the $$N$$- and $$M$$-dimensional Euclidean spaces, respectively, find a vector $$x^*$$ for which $$x^*\in C$$ and $$Ax^*\in Q$$, where $$C$$ is the intersection of the sets $$C_i$$, $$i= 1,\dots, t$$ and $$Q$$ is the intersection of $$Q_j$$, $$j= 1,\dots, r$$, and $$A$$ is a given $$M\times N$$ real matrix.
A projection algorithm for solving this problem that minimizes a proximity function that measures the distance from a point from all sets, is proposed as well as its generalization, in which the Bregman distances are used. Application of the method to inverse problems of intensity-modulated radiation therapy treatment planning is briefly described.

##### MSC:
 65K05 Numerical mathematical programming methods 90C25 Convex programming 92C50 Medical applications (general)
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