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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.

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.

Reviewer: Karel Zimmermann (Praha)

### MSC:

65K05 | Numerical mathematical programming methods |

90C25 | Convex programming |

92C50 | Medical applications (general) |