The relaxed inexact projection methods for the split feasibility problem

Abstract

In this paper we present several relaxed inexact projection methods for the split feasibility problem (SFP). Each iteration of the first proposed algorithm consists of a projection onto a halfspace containing the given closed convex set. The algorithm can be implemented easily and its global convergence to the solution can be established under suitable conditions. Moreover,we present some modifications of the relaxed inexact projection method with constant stepsize by adopting Armijo-like search. We furthermore present a variable-step relaxed inexact projection method which does not require the computation of the matrix inverses and the largest eigenvalue of the matrix ATA, and the objective function can decrease sufficiently at each iteration. We show convergence of these modified algorithms under mild conditions. Finally, we perform some numerical experiments, which show the behavior of the algorithms proposed.