Bounded perturbation resilience of extragradient-type methods and their applications
Abstract In this paper we study the bounded perturbation resilience of the extragradient and the subgradient extragradient methods for solving a variational inequality (VI) problem in real Hilbert spaces. This is an important property of algorithms which guarantees the convergence of the scheme unde...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2017-11-01
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| Series: | Journal of Inequalities and Applications |
| Subjects: | |
| Online Access: | http://link.springer.com/article/10.1186/s13660-017-1555-0 |
| _version_ | 1811331289193644032 |
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| author | Q-L Dong A Gibali D Jiang Y Tang |
| author_facet | Q-L Dong A Gibali D Jiang Y Tang |
| author_sort | Q-L Dong |
| collection | DOAJ |
| description | Abstract In this paper we study the bounded perturbation resilience of the extragradient and the subgradient extragradient methods for solving a variational inequality (VI) problem in real Hilbert spaces. This is an important property of algorithms which guarantees the convergence of the scheme under summable errors, meaning that an inexact version of the methods can also be considered. Moreover, once an algorithm is proved to be bounded perturbation resilience, superiorization can be used, and this allows flexibility in choosing the bounded perturbations in order to obtain a superior solution, as well explained in the paper. We also discuss some inertial extragradient methods. Under mild and standard assumptions of monotonicity and Lipschitz continuity of the VI’s associated mapping, convergence of the perturbed extragradient and subgradient extragradient methods is proved. In addition we show that the perturbed algorithms converge at the rate of O ( 1 / t ) $O(1/t)$ . Numerical illustrations are given to demonstrate the performances of the algorithms. |
| first_indexed | 2024-04-13T16:18:29Z |
| format | Article |
| id | doaj.art-c28f961c82d9490da5300ed6330dbdd2 |
| institution | Directory Open Access Journal |
| issn | 1029-242X |
| language | English |
| last_indexed | 2024-04-13T16:18:29Z |
| publishDate | 2017-11-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of Inequalities and Applications |
| spelling | doaj.art-c28f961c82d9490da5300ed6330dbdd22022-12-22T02:40:00ZengSpringerOpenJournal of Inequalities and Applications1029-242X2017-11-012017112810.1186/s13660-017-1555-0Bounded perturbation resilience of extragradient-type methods and their applicationsQ-L Dong0A Gibali1D Jiang2Y Tang3Tianjin Key Laboratory for Advanced Signal Processing, College of Science, Civil Aviation University of ChinaDepartment of Mathematics, ORT Braude CollegeTianjin Key Laboratory for Advanced Signal Processing, College of Science, Civil Aviation University of ChinaDepartment of Mathematics, NanChang UniversityAbstract In this paper we study the bounded perturbation resilience of the extragradient and the subgradient extragradient methods for solving a variational inequality (VI) problem in real Hilbert spaces. This is an important property of algorithms which guarantees the convergence of the scheme under summable errors, meaning that an inexact version of the methods can also be considered. Moreover, once an algorithm is proved to be bounded perturbation resilience, superiorization can be used, and this allows flexibility in choosing the bounded perturbations in order to obtain a superior solution, as well explained in the paper. We also discuss some inertial extragradient methods. Under mild and standard assumptions of monotonicity and Lipschitz continuity of the VI’s associated mapping, convergence of the perturbed extragradient and subgradient extragradient methods is proved. In addition we show that the perturbed algorithms converge at the rate of O ( 1 / t ) $O(1/t)$ . Numerical illustrations are given to demonstrate the performances of the algorithms.http://link.springer.com/article/10.1186/s13660-017-1555-0inertial-type methodbounded perturbation resilienceextragradient methodsubgradient extragradient methodvariational inequality |
| spellingShingle | Q-L Dong A Gibali D Jiang Y Tang Bounded perturbation resilience of extragradient-type methods and their applications Journal of Inequalities and Applications inertial-type method bounded perturbation resilience extragradient method subgradient extragradient method variational inequality |
| title | Bounded perturbation resilience of extragradient-type methods and their applications |
| title_full | Bounded perturbation resilience of extragradient-type methods and their applications |
| title_fullStr | Bounded perturbation resilience of extragradient-type methods and their applications |
| title_full_unstemmed | Bounded perturbation resilience of extragradient-type methods and their applications |
| title_short | Bounded perturbation resilience of extragradient-type methods and their applications |
| title_sort | bounded perturbation resilience of extragradient type methods and their applications |
| topic | inertial-type method bounded perturbation resilience extragradient method subgradient extragradient method variational inequality |
| url | http://link.springer.com/article/10.1186/s13660-017-1555-0 |
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