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

Abstract

Given nonempty closed convex subsets $C_i\subseteq R^m$, $i=1,2,\dots ,t$ and nonempty closed convex subsets $Q_j\subseteq R^n$, $j=1,2,\dots ,r$, in the $n$- and $m$-dimensional Euclidean spaces, respectively. The multiple-set split feasibility problem (MSSFP) proposed by Censor is to find a vector $x\in {\bigcap }_{i=\mathrm{1}}^t{C}_i$ such that $Ax\in {\bigcap }_{j=\mathrm{1}}^r{Q}_j$, where $A$ is a given $M\times N$ 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.