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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 N , i=1,,t and closed convex sets Q j M , j=1,,r, in the N- and M-dimensional Euclidean spaces, respectively, find a vector x * for which x * C and Ax * Q, where C is the intersection of the sets C i , i=1,,t and Q is the intersection of Q j , j=1,,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:
65K05Mathematical programming (numerical methods)
90C25Convex programming
92C50Medical applications of mathematical biology