Relaxation Sequence Projection Algorithm And Its Application For Solving The Problem Of Set - Splitting Probability

The split feasible problem(SFP) is an important research problem in the field of optimization, and the multiple-sets split feasibility problem(MSFP), as an extension problem of the split feasibility problem, was put forward by Censor in 2005. The multiple-sets split feasibility problem is required to find a point in the intersection of a family of nonempty closed convex subsets in one space such that its image under a linear transformation is in the intersection of another family of nonempty closed convex subsets in the image space. It has been used in signal processing, image reconstruction, and the intensity- modulated radiation therapy(IMRT). It has been concerned widely by many domestic and foreign experts and scholars since being proposed. Some fruitful results have been achieved in its theories and algorithms. But many of the existing algorithms have to calculate either the orthogonal projections onto the nonempty closed convex sets or the suitable step size by computing()TρA A 、estimating Lipschitz coefficient or doing line research, these two cases are often difficult to implement in practice. In 2014, Liu and Qu proposed a step size which can be calculated directly. This greatly reduced the amount of computation. Subsequently, Liu and Qu introduced a sequence projection algorithm. However, it involves the projections onto the closed convex sets which is very difficult to achieve in practice. In this paper we relax the closed convex sets by constructing the half space to improve this insufficiency. We propose the relaxed sequence projection algorithm which is easier in calculating. The full text have four chapters, and the structure is as follows:The first chapter is an introduction. We state the origin of the multiple-sets split feasibility problem, the application background of it and its research status. We also describe the main work of this paper.In the second chapter, we solve the split feasibility problem with the 1-norm constraint as a special case of the multiple-sets split feasibility problem. We get the alternating projections algorithm based on sequence projection algorithm and we use it to solve the split feasibility problem with the 1- norm constraint. However, this algorithm involves the projection onto the closed convex sets which is very difficult to achieved in practice. In the latter part of this chapter, a modification algorithm called relaxed alternating projections algorithm is given. Besides, we prove that the iteration point sequence generated by this algorithm converges to a solution of the split feasibility problem with 1- norm constraint.In the third chapter, due to the sequence projection algorithm for solving the multiple-sets split feasibility problem involves the projection onto the closed convex set, we improve it by constructing the half space. A modification algorithm called relaxed sequence projection algorithm is given. Besides, we prove that the iteration point sequence generated by this algorithm converges to a solution of the multiple-sets split feasibility problem.In the fourth chapter,based on the relaxed sequence projection algorithm, we unify, generalize, and review some related algorithms. And we get the solution of the split feasibility problem with 2- normal constraint.