Study Of The Relaxed Projection Algorithm For The Split Feasibility Problem

Optimization method is an important part of operations resarch. It has wide applications to many fields, such as natural science, social science, practical production, engieering design and modern management, et. This thesis includes four chapters, which mainly discusses the relaxed projection algorithms for solving the split feasibility problem.Chapter 1 is the introduction. We describe the research situations of the split feasibility problem, the main results obtained in this thesis and preliminary knowledge.In chapter 2, we present a new halfspace-relaxation projection method for the split feasibility problem. For this problem, Qu(2005)proposed an extragra-dient algorithm. Based on above algorithm, we get relaxed projection algorithm by weakening the predictor stepsize in predictor step and getting the optimal corrector stepsize in corrector step. At last, we obtain the global convergence for this algorithm for the case where the solution set of the split feasibility problem is nonempty and give the preliminary numerical experiments which show that the improved algorithm has good behavior.In chapter 3, we present one projection algorithm for the split feasibility problem. Based on the algorithm proposed in Qu(2008) and the method presented in the second chapter, we obtain it by a new reformulation for the split feasibility problem. The objective function in the new reformulation is convexly quadratic. The corresponding gradient and Hessian matrix can be computed very easily. At each iteration, it needs only to compute one-projection onto a halfspace containing the closed convex set so that the convergence speeds up. Global convergence of the algorithm is shown. In addition, preliminary computational experience is also reported. In Chapter 4, we present a relaxed projection algorithm for the mutiple-sets split feasibility problem. For this problem, Y air Censor,Avi Motova,Alexander Segal(2006) proposed a projection algorithm which possesses good global convergence property. In this chapter, we modify the algorithm from the following two aspects: (1) We get iterative stepsize by like-Armijo search and avoid the defect of the fixed stepsize in the original algorithm; (2)This algorithm needs only to compute the projection onto the halfspace instead onto the closed convex set, which is implemented very easily. It has full convergence to the solution for the case where the solution set of muliple-sets split feasibility problem is nonempty. At last, numerical examples are given to show the effectiveness of this algorithm.