Self-Adaptive and Relaxed Self-Adaptive Projection Methods for Solving the Multiple-Set Split Feasibility Problem
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 man...
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| Format: | Article |
| Language: | English |
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Wiley
2012-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2012/958040 |
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| author | Ying Chen Yuansheng Guo Yanrong Yu Rudong Chen |
| author_facet | Ying Chen Yuansheng Guo Yanrong Yu Rudong Chen |
| author_sort | Ying Chen |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-b812a9c2c05f4dc8aa7e596a1e3ad10d |
| institution | OA Journals |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2012-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-b812a9c2c05f4dc8aa7e596a1e3ad10d2025-08-20T02:19:11ZengWileyAbstract and Applied Analysis1085-33751687-04092012-01-01201210.1155/2012/958040958040Self-Adaptive and Relaxed Self-Adaptive Projection Methods for Solving the Multiple-Set Split Feasibility ProblemYing Chen0Yuansheng Guo1Yanrong Yu2Rudong Chen3Textile Division, Tianjin Polytechnic University, Tianjin 300160, ChinaDepartment of Mathematics, Tianjin Polytechnic University, Tianjin 300160, ChinaDepartment of Mathematics, Tianjin Polytechnic University, Tianjin 300160, ChinaDepartment of Mathematics, Tianjin Polytechnic University, Tianjin 300160, ChinaGiven 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.http://dx.doi.org/10.1155/2012/958040 |
| spellingShingle | Ying Chen Yuansheng Guo Yanrong Yu Rudong Chen Self-Adaptive and Relaxed Self-Adaptive Projection Methods for Solving the Multiple-Set Split Feasibility Problem Abstract and Applied Analysis |
| title | Self-Adaptive and Relaxed Self-Adaptive Projection Methods for Solving the Multiple-Set Split Feasibility Problem |
| title_full | Self-Adaptive and Relaxed Self-Adaptive Projection Methods for Solving the Multiple-Set Split Feasibility Problem |
| title_fullStr | Self-Adaptive and Relaxed Self-Adaptive Projection Methods for Solving the Multiple-Set Split Feasibility Problem |
| title_full_unstemmed | Self-Adaptive and Relaxed Self-Adaptive Projection Methods for Solving the Multiple-Set Split Feasibility Problem |
| title_short | Self-Adaptive and Relaxed Self-Adaptive Projection Methods for Solving the Multiple-Set Split Feasibility Problem |
| title_sort | self adaptive and relaxed self adaptive projection methods for solving the multiple set split feasibility problem |
| url | http://dx.doi.org/10.1155/2012/958040 |
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