Self-adaptive Algorithms For The Split Feasibility Problems

Since the split feasibility problems and multiple-sets split feasibility problems have wide applications such as radiation therapy,image reconstruction and signal processing,research on the algorithms of these problems is of both theoretical and practical importance.The topic has been intensively and extensively studied and developed.Many nice results have been obtained.Based on the classical CQ algorithm,this paper is devoted to propose some modified algorithms with self-adaptive stepsize.The main content is listed below.First,we consider the split feasibility problems.Based on the ball-relaxed CQ algorithm with self-adaptive stepsize,inertial term is imposed to accelerate the convergence rate of the algorithm;Halpern iterative scheme is used to ensure the strong convergence in the infinite dimensional Hilbert space.Second,again based on the ball-relaxed CQ algorithm,a modified CQ algorithm is proposed to solve the split feasibility problems.The stepsize of the algorithm is determined by the Armijo-line search.The weak convergence is proved in the infinite dimensional Hilbert space.Third,we consider the multiple-sets split feasibility problems.Two relaxed CQ algorithms with self-adaptive stepsize and inertial term are proposed.One is mixed cyclic/simultaneous,and the other is cyclic.The weak convergence of the algorithms is verified in the infinite dimensional Hilbert space.