Gradient Projection-like Algorithms For Split Feasible Problems With Sparse Constraints

The split feasibility problem with sparse constraints is a very important field in optimization.Sparse constraints mean the most elements of a variable are zero,or the number of zero components of a variable is more than a given value.Split feasibility problem arises from engineering practice,and it has been widely used in signal processing,compressive sensing,image processing,machine learning,and pattern recognition and other aspects.Recently,with the development of computer technology,more and more data are obtained in scientific research,the space and time needed to process these data are also increasing,so the research on sparsity appears.For this reason,many traditional algorithms are difficult to solve the split feasibility problem with sparse constraints.Therefore,to design effective algorithm for solving the split feasibility problem with sparse constraints have important theoretical significance and practical value,and is one of the current research topics.This paper is mainly divided into three chapters.The main structure is shown below:The first part is introduction.We mainly introduced the definition,the research situation of the split feasibility problem with sparse constraints and the main work of this paper.In the second chapter,we present the gradient-projection algorithm,which applies a nonmonotonic Armijo step search to determine the step size.In this algorithm,the step size of each iteration is relatively large,even if the objective function does not necessarily decline at every step,but on the whole it is declining.And we proved the convergence of this algorithm.Finally,numerical examples are given to demonstrate the feasibility and effectiveness of the algorithm.The third chapter gives the gradient-projection quasi-Newton algorithm with Wolfe steplength.In this algorithm,we only need to consider the first-order gradient of the objective function,and the second-order differentiability of the objective function and the s-regularity of the matrix are not involved.And we proved the convergence of the algorithm.Finally,the efficiency and feasibility of the proposed algorithm is verified by numerical examples.