The Improved Valiant Projection Algorithm And Halpern Algorithm In Hilbert Spaces

In this dissertation,two projection algorithms are proposed to solve the convex feasibility problem.One is the improved Valiant projection algorithm,and the other is the improved Halpern algorithm.In essence,the improved Valiant projection algorithm transforms the multi-set convex feasibility problem into the feasibility problem of two sets with the help of the thought of Valiant projection and product space technology,that is,it is an algorithm to solve the multi-set convex feasibility problem.Based on the classical Halpern algorithm,the improved Halpern algorithm generalizes the fixed point to the moving point and gives the strong convergence result.Compared with the classical case,the improved Halpern algorithm has a wider range of applications.The dissertation consists of five parts and the structure is as follows.In chapter 1,we introduce the research background of projection algorithm in Hilbert space at home and abroad and the main work of this dissertation.In chapter 2,elaborates the preliminary knowledge needed in this dissertation.In chapter 3,an improved Valiant projection algorithm is proposed to prove its weak convergence.When projection on a closed convex set is difficult to calculate,we transform multiple set convex feasibility problem into the feasibility of the two collections with the help of alternating projection of Valiant put forward by the Censor and product space technology.This makes the algorithm more effective than classical of alternating projection algorithm.In chapter 4,an improved Halpern projection algorithm is studied and its strong convergence is proved.That is,a single projection is applied to a given set to calculate the projection on the intersection of a finite number of closed convex sets,and the result of strong convergence is given.In chapter 5,the full dissertation summary,explains the main content of this dissertation and the future trends.