Self-Adaptive and Relaxed Self-Adaptive Projection Methods for Solving the Multiple-Set Split Feasibility Problem

Given nonempty closed convex subsets C-i subset of R-m, i = 1, 2,..., t and nonempty closed convex subsets Q(j) subset of R-n, j = 1, 2,..., r, in the n- and m-dimensional Euclidean spaces, respectively. The multiple-set split feasibility problem (MSSFP) proposed by Censor is to find a vector x is an element of boolean AND(i)(t=1) C-i such that Ax is an element of boolean AND(r)(j=1) Q(j), where A is a given M x N real matrix. It serves as 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. MSSFP has a variety of specific applications in real world, such as medical care, image reconstruction, and signal processing. In this paper, for the MSSFP, we first propose a new self-adaptive projection method by adopting Armijo-like searches, which dose not require estimating the Lipschitz constant and calculating the largest eigenvalue of the matrix A(T)A; besides, it makes a sufficient decrease of the objective function at each iteration. Then we introduce a relaxed self-adaptive projection method by using projections onto half-spaces instead of those onto convex sets. Obviously, the latter are easy to implement. Global convergence for both methods is proved under a suitable condition.