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

Abstract

Given nonempty closed convex subsets Ci⊆Rm, i = 1, 2, …, t and nonempty closed convex subsets Qj⊆Rn, 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 such that , where A is a given M × 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 ATA; 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.