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Abstract and Applied Analysis
Volume 2012, Article ID 958040, 11 pages
Research Article

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

1Textile Division, Tianjin Polytechnic University, Tianjin 300160, China
2Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China

Received 10 October 2012; Accepted 17 November 2012

Academic Editor: Yongfu Su

Copyright © 2012 Ying Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Given nonempty closed convex subsets , and nonempty closed convex subsets , , in the - and -dimensional Euclidean spaces, respectively. The multiple-set split feasibility problem (MSSFP) proposed by Censor is to find a vector such that , where is a given 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 ; 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.