| The split feasibility problem(SFP)is a kind of mathematical problem with wide applica-tion.It has important applications in decision theory,biological engineering,control theory,military,and the rapidly developing fast image processing technology,language processing system,remote sensing and other fields,which has attracted the attention of many scholars.Scholars have studied the split feasibility problem from various perspectives and proposed many algorithms,but the study on the unconstrained optimization form of this problem is few.Since the gradient-based algorithm has the advantages of fast iteration speed and short computation time,this paper designs a series of gradient-based algorithms for split feasibility problem from the perspective of unconstrained optimization.The structure is as follows:The first chapter is the introduction,which mainly introduces the basic concepts,research status and application of the split feasibility problem,and gives the main work of this paper.The second chapter is the preliminary knowledge,which introduces the basic definitions and lemmas related to this paper.In the third chapter,a class of gradient-based algorithm for solving the split feasibility problem has been designed.Firstly,the split feasibility problem is transformed into an unconstrained optimization form.Then a new step size α_k is selected by four different methods.We simplify the iterative steps of the algorithm and prove the global convergence of the algorithm.Finally,the effectiveness of the algorithm is verified by numerical experiments.In the fourth chapter,we propose a class of gradient algorithm with relaxed projection.By constructing two half-spaces to relax the closed convex set C,Q,we avoid the situation that the projection onto C and Q is difficult to calculate.In addition,we don’t need to calculate the inverse and maximum eigenvalue of the matrix,which saves the calculation time and improves the efficiency of the algorithm.The global convergence of the algorithm is proved.Finally the feasibility and effectiveness of the algorithm is verified by numerical experiments.The fifth chapter summarizes the research content of this paper and puts forward the direction of further research. |
