Gradient Projection Algorithm For Non-convex Splitting Feasible Proble

Optimization problem is a significant research field in mathematics.It is used in financial decision-making,engineering design,economic management,transportation planning,defense technology and so on.Among them,the Split Feasibility Problem(SFP)is a critical and common problem,which is widely applied to signal processing,image restoration,intensity modulated radiation therapy and other fields.With the development of society and the need of practice,the application of split feasibility problem in various fields is no longer limited to convex sets,that is,the and are non-convex sets.As we all know,there are many algorithms to solve the split feasibility problem,such as CQ algorithm,relaxed CQ algorithm and DR algorithm.However,the algorithms on non-convex cases are few.Such problem,which is called as non-convex split feasibility problem,needs to be solved urgently.Therefore,the theoretical research and algorithm design of it have important practical significance.This paper is divided into five chapters,and the structure is as follows:The first part is introduction,which mainly introduces the concepts,research situation of non-convex split feasibility problem and the main work of our paper.The second chapter is preliminaries,focusing on the relevant definitions of non-convex function,KL property and uniformized KL property.In the third chapter,a gradient projection algorithm,which employs an Armijo-type line search to determine the step size,is presented.In this algorithm,the step size is no longer related to the Lipschitz constant of the gradient of the objective function,while the descent of the objective function is guaranteed.The convergence of the algorithm is proved.Finally,the feasibility and effectiveness of the proposed algorithm are verified by numerical examples.The fourth chapter presents an inertial forward-reflected-backward splitting algorithm.This algorithm uses the line search to obtain the step size and combines the inertial method,which can accelerate the convergence rate of the algorithm.We prove the convergence and convergence rate of the algorithm.Finally,numerical examples are given to verify the feasibility and effectiveness of the algorithm.The fifth chapter is conclusions.We end this paper by summarizing the present work and providing some directions for future research.