Solving the Variational Inequality Problem Defined on Intersection of Finite Level Sets
Consider the variational inequality VI(C,F) of finding a point x*∈C satisfying the property 〈Fx*,x-x*〉≥0, for all x∈C, where C is the intersection of finite level sets of convex functions defined on a real Hilbert space H and F:H→H is an L-Lipschitzian and η-strongly monotone operator. Relaxed and s...
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| Format: | Article |
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Wiley
2013-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2013/942315 |
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| author | Songnian He Caiping Yang |
| author_facet | Songnian He Caiping Yang |
| author_sort | Songnian He |
| collection | DOAJ |
| description | Consider the variational inequality VI(C,F) of finding a point x*∈C satisfying the property 〈Fx*,x-x*〉≥0, for all x∈C, where C is the intersection of finite level sets of convex functions defined on a real Hilbert space H and F:H→H is an L-Lipschitzian and η-strongly monotone operator. Relaxed and self-adaptive iterative algorithms are devised for computing the unique solution of VI(C,F). Since our algorithm avoids calculating the projection PC (calculating PC by computing several sequences of projections onto half-spaces containing the original domain C) directly and has no need to know any information of the constants L and η, the implementation of our algorithm is very easy. To prove strong convergence of our algorithms, a new lemma is established, which can be used as a fundamental tool for solving some nonlinear problems. |
| format | Article |
| id | doaj-art-19ee71659b66470c9b834f0bf09bf3ee |
| institution | DOAJ |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2013-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-19ee71659b66470c9b834f0bf09bf3ee2025-08-20T03:17:52ZengWileyAbstract and Applied Analysis1085-33751687-04092013-01-01201310.1155/2013/942315942315Solving the Variational Inequality Problem Defined on Intersection of Finite Level SetsSongnian He0Caiping Yang1College of Science, Civil Aviation University of China, Tianjin 30030, ChinaCollege of Science, Civil Aviation University of China, Tianjin 30030, ChinaConsider the variational inequality VI(C,F) of finding a point x*∈C satisfying the property 〈Fx*,x-x*〉≥0, for all x∈C, where C is the intersection of finite level sets of convex functions defined on a real Hilbert space H and F:H→H is an L-Lipschitzian and η-strongly monotone operator. Relaxed and self-adaptive iterative algorithms are devised for computing the unique solution of VI(C,F). Since our algorithm avoids calculating the projection PC (calculating PC by computing several sequences of projections onto half-spaces containing the original domain C) directly and has no need to know any information of the constants L and η, the implementation of our algorithm is very easy. To prove strong convergence of our algorithms, a new lemma is established, which can be used as a fundamental tool for solving some nonlinear problems.http://dx.doi.org/10.1155/2013/942315 |
| spellingShingle | Songnian He Caiping Yang Solving the Variational Inequality Problem Defined on Intersection of Finite Level Sets Abstract and Applied Analysis |
| title | Solving the Variational Inequality Problem Defined on Intersection of Finite Level Sets |
| title_full | Solving the Variational Inequality Problem Defined on Intersection of Finite Level Sets |
| title_fullStr | Solving the Variational Inequality Problem Defined on Intersection of Finite Level Sets |
| title_full_unstemmed | Solving the Variational Inequality Problem Defined on Intersection of Finite Level Sets |
| title_short | Solving the Variational Inequality Problem Defined on Intersection of Finite Level Sets |
| title_sort | solving the variational inequality problem defined on intersection of finite level sets |
| url | http://dx.doi.org/10.1155/2013/942315 |
| work_keys_str_mv | AT songnianhe solvingthevariationalinequalityproblemdefinedonintersectionoffinitelevelsets AT caipingyang solvingthevariationalinequalityproblemdefinedonintersectionoffinitelevelsets |