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 , and closed convex sets , , in the - and -dimensional Euclidean spaces, respectively, find a vector for which and , where is the intersection of the sets , and is the intersection of , , and is a given 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.