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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 $C_i\subseteq\bbfR^N$, $i= 1,\dots, t$ and closed convex sets $Q_j\subseteq\bbfR^M$, $j= 1,\dots, r$, 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.

65K05Mathematical programming (numerical methods)
90C25Convex programming
92C50Medical applications of mathematical biology
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