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2012 Self-Adaptive and Relaxed Self-Adaptive Projection Methods for Solving the Multiple-Set Split Feasibility Problem
Ying Chen, Yuansheng Guo, Yanrong Yu, Rudong Chen
Abstr. Appl. Anal. 2012: 1-11 (2012). DOI: 10.1155/2012/958040

Abstract

Given nonempty closed convex subsets C i R m , i = 1 , 2 , , t and nonempty closed convex subsets Q j 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 i = 1 t C i such that A x j = 1 r Q j , 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 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.

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Ying Chen. Yuansheng Guo. Yanrong Yu. Rudong Chen. "Self-Adaptive and Relaxed Self-Adaptive Projection Methods for Solving the Multiple-Set Split Feasibility Problem." Abstr. Appl. Anal. 2012 1 - 11, 2012. https://doi.org/10.1155/2012/958040

Information

Published: 2012
First available in Project Euclid: 28 March 2013

zbMATH: 1263.90057
MathSciNet: MR2999940
Digital Object Identifier: 10.1155/2012/958040

Rights: Copyright © 2012 Hindawi

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