A self-adaptive projection method for solving the multiple-sets split feasibility problem

The multiple-sets split feasibility problem, a generalization and extension of the split feasibility problem, has a variety of specific applications in real world, such as medical care, image reconstruction and signal processing. It can be a model for many inverse problems where constraints are imposed on the solutions in the domain of a linear operator as well as in the operator's range. Censor et al (2005 Inverse Problems 21 2171–84) proposed a method for solving the multiple-sets split feasibility problem, whose efficiency depends heavily on step size, a fixed constant related to the Lipschitz constant of ∇p(x) (see the definition in section 1). To estimate the Lipschitz constant is a very difficult, if not an impossible task. On the other hand, even if we know the Lipschitz constant, a method with fixed step size may be slow. In this paper, we propose a new method for solving the multiple-sets split feasibility problem by adopting variable step sizes, which chooses suitable step sizes self-adaptively, based on the information from the current iterate. It thus avoids the difficult task of estimating the Lipschitz constant, while the efficiency is enhanced greatly. We prove the global convergence of the new method and report our numerical results, which are promising.