Projection Gradient Algorithm For Sparse Splitting Feasible Problems

The sparse split feasibility problem refers to a split feasibility problem with sparse con-straints.A sparse constraint means that most elements of a variable are zero,or the number of non-zero components of a variable does not exceed a given value.Split feasibility problem arises from engineering practice and is a very important type of problem in optimization problems.These problems are widely used in regression analysis,compressed sensing,pat-tern recognition,machine learning and other fields.In recent years,with the application of compressed sensing technology in image restoration,signal processing,etc.,variables are required to be sparse.Due to the sparse property of variables,the sparse split feasibili-ty problem cannot be solved by many traditional algorithms.So studying the sparse split feasibility problem is meaningful for us.This paper is divided into four chapters.The structure of this paper is organized as follows:In Chapter 1,we mainly introduce the definition and the research situation of the sparse split feasibility problem and main work of this paper.In Chapter 2,we propose a projected gradient quasi-Newton algorithm to solve the s-parse split feasibility problem.Under this algorithm,it is not necessary to consider the s-regularity of the matrix and the second-order differentiability of the objective function,so the calculation of Hesse matrix of objective function is avoided.We prove that the se-quence of iterates generated by this algorithm can converge to an ?-stationary point of the sparse split feasibility problem.Finally,the efficiency of the proposed algorithm is verified by numerical examples.In Chapter 3,we present projected gradient algorithm with a new step size rule for solving the sparse split feasibility problem.Under this step size rule,The objective function can be reduced sufficiently at each step.It is proved that the sequence of iterates generated by this algorithm can converge to a solution of the sparse split feasibility problem.Finally,the feasibility of the proposed algorithm is verified by numerical examples.In Chapter 4,we summarize the research content of this paper and put forward the direction for further research.