Strong convergence of Bregman projection algorithms for solving split feasibility problems

Abstract

Bregman distance methods play a key role in solving problems in nonlinear analysis and optimization theory, since the Bregman distance is a useful substitute for the metric. The main purpose of this paper is to investigate two new iterative algorithms based on the Bregman distance and the Bregman projection for solving split feasibility problems in real Hilbert spaces. The algorithms are constructed around these methods: Byrne’s CQ method, Polyak’s gradient method, Halpern method, and hybrid projection method. The proposed methods involve inertial extrapolation terms and self-adaptive step sizes. We prove that the proposed iterations converge strongly to the Bregman projection of the initial point onto the solution set. Some numerical examples are provided to illustrate the computational effectiveness of our algorithms. The main results extend and improve the recent results related to the split feasibility problem.