The Minimum Norm Solution To The Split Feasible Problem

Split feasibility problem(SFP)is a very common and important problem in the optimization field.It has been widely used in medical,signal processing,image reconstruction,compressed sensing and other engineering fields.Since this problem was proposed,many scholars from home and abroad have analyzed and studied it.They put forward many algorithms about this problem.With the further research on the split feasibility problem,experts and scholars have studied split feasibility problem and its related optimization problems from different perspectives.The minimum norm solution of split feasibility problem is one of them.At present,the research about this problem is relatively less.Therefore,this paper makes a further study on the minimum norm solution of split feasibility problem.This paper consists of four chapters.The structure of this paper is as follows:In Chapter 1,we mainly state the origin of the split feasibility problem,the application background,research status and main works of this paper.In Chapter 2,we present a gradient algorithm for solving the minimum 2-norm solution of the split feasibility problem.The problem which is the minimum 2-norm solution of the split feasibility problem is transformed into an unconstrained optimization problem and solved by the gradient algorithm.We expand the selection range of step size,weaken the convergence condition of the algorithm,and prove the convergence of the algorithm.Finally,the practicability of the algorithm is verified by numerical experiments.In Chapter 3,we research the minimum 1-norm solution of the split feasibility problem.We use the idea of 1-norm regularization to establish the relationship between the solution of unconstrained optimization problem and the minimum 1-norm solution of split feasibility problem.We get the conclusion that any accumulation point of the solution of unconstrained optimization problem is the minimum 1-norm solution of split feasibility problem.At last,we give an application of the minimum 1-norm solution of the split feasibility problem.In Chapter 4,we summarize the research content of this paper,and put forward the direction of further research.