Conjugate Gradient Algorithm For Sparse Splitting Feasible Problems

The sparse split feasibility problem refers to a split feasibility problem with sparse constraints.It has been widely used in signal processing,image restoration,compressed sensing,machine learning and other fields.In recent years,more and more scholars have paid attention to it,and some theoretical analysis and algorithmic research on this problem have been put forward one after another.However,due to the sparse nature of variables,many traditionaliterative algorithms cannot solve the sparse split feasibility problem.So,it is a meaningful research to design an algorithm for sparse split feasibility problem.In addition to abstract,the full text includes four parts.The main structure is as follows:The first chapter is the introduction,which mainly elaborates the definition,research significance and research status of the sparse split feasibility problem.On this basis,the development status,existing problems and the main work of this paper are introduced.In Chapter 2,by using the equivalence between the minimum 0-norm solution and the minimum 1-norm solution of the split feasibility problem,we transformed the split feasibility problem into the problem of solving the 1-norm solution.We transformed the problem into a non-smooth unconstrained convex optimization problem,and then used Nesterov smoothing technique to obtain a smooth unconstrained convex optimization problem.Then we proposed a modified PRP conjugate gradient algorithm,which used to solve the sparse split feasibility problem.And the convergence of the algorithm is proved.The numerical results verified the effectiveness of the algorithm.In Chapter 3,we designed a new modified CD conjugate gradient algorithm with generalized Armijo step size rule to solve the sparse split feasibility problem,and proved that the algorithm has global convergence.Finally,numerical examples were given to illustrate theeffectiveness of the algorithm.Chapter 4 is a summary of the whole paper and a prospect of future research directions.