On The Relaxed Projection Algorithm And Its Promotion For The Split Feasibility Problem

In this dissertation, we mainly investigate the algorithms of the Split Fea-siblity Problem, the mutiple-sets split feasibility problem and the split common fixed point problem. We design three methods in solving these problems in this paper. The dissertation has four chapters.Chapter 1 is the introduction. We describe the research situations of the split feasibility problem, the main results obtained in this thesis.In chapter 2, we present a general relaxation projection algorithm for the split feasibility problem. This algorithm employs projections onto hyperplanes that separate "small" ball around current iterate point from the feasible set of the SFP instead of projections onto the feasible set itself. Our algorithmic scheme includes the classical projection method and subgradient projection method as special cases. At last, we obtain the global convergence for this algorithm and numerical examples are given to show the effectiveness of this algorithm.In chapter 3, we present a relaxed projection algorithm for the mutiple-sets split feasibility problem. The mutiple-sets split feasibility problem generalizes the split feasibility problem. In this chapter, by casting this problem into the split feasibility problem in a suitable product space we are able to present a projection algorithm that generates convergent. In addition, preliminary computational experience is also reported.In chapter 4, we propose the split common fixed point problem that requires to find a common fixed point of a family of operators in one space whose image under a linear transformation is a common fixed point of another family of op-erators in the image space. This generalizes the convex feasibility problem, the two-sets split feasibility problem and the multiple sets split feasibility problem. In this chapter, we present an algorithm for solving this split common fixed point problem by a class of directed operators. The directed operator includes the pro-jections used in Chapter 2 and Chapter 3 as special cases. At last, we obtain the global convergence for this algorithm.