Theory And Algorithm Study For One Type Of Feasibility Problem

In this thesis,we mainly consider the linear feasibility problem,the convex feasibility problem respectively.The paper consists of four chapters.Chapter 1 gives an introduction of the thesis,which mainly discusses the current development of the discussed issue,i.e.the linear feasibility problem,the convex feasibility problem and furthermore,the main contribution of this paper is also listed in this chapter.In chapter 2,basing on the existing projection method for solving the linear feasibility problem,we present a new accelerated projection method by adopting a accelerated procedure and a surrogate technique.For this new method,we first prove its global convergence provided that the solution set is nonempty and then show that it is R-linearly convergent under general situation.The given preliminary computational experiments show that our modified method is efficient.In chapter 3,we present a new projection method for solving the linear feasibility problem based on the method proposed by Dudek(2007).Compared with Dudek's iterative method of the line search procedure with fixed stepsize,our method adopts a surrogate technique to obtain the new iterative point to avoid the phenomenon of the unbounded stepsize in the iterative procedure of method. Compared with the existing method in the previous chapter,we first struct a new half-space by using the current point and the previous iteration point,then surrogate the projecting on the new half-space by the surrogate technique to shorten the distance between the new iterative point and the solution.For this new method,we first prove its global convergence provided that the solution set is nonempty.The given preliminary computational experiments show that this method has a good performance. In chapter 4,we propose a new projection method for solving the convex feasibility problem based on the method proposed by Eremin(1970).Compared with the iterative method of Eremin,the new method adopts two projections and a surrogate technique to obtain the new iterative point.For this new method, we first prove its global convergence provided that the solution set is nonempty and then show that it is R-linearly convergent under general situation.The given preliminary computational experiments show that our method is promising.