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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 ${C}_{i}\subseteq {ℝ}^{N}$, $i=1,\cdots ,t$ and closed convex sets ${Q}_{j}\subseteq {ℝ}^{M}$, $j=1,\cdots ,r$, 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 $M×N$ 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