*(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 ${C}_{i}\subseteq {\mathbb{R}}^{N}$, $i=1,\cdots ,t$ and closed convex sets ${Q}_{j}\subseteq {\mathbb{R}}^{M}$, $j=1,\cdots ,r$, in the $N$- and $M$-dimensional Euclidean spaces, respectively, find a vector ${x}^{*}$ for which ${x}^{*}\in C$ and $A{x}^{*}\in Q$, where $C$ is the intersection of the sets ${C}_{i}$, $i=1,\cdots ,t$ and $Q$ is the intersection of ${Q}_{j}$, $j=1,\cdots ,r$, 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 | Mathematical programming (numerical methods) |

90C25 | Convex programming |

92C50 | Medical applications of mathematical biology |