A New Gap Function for Vector Variational Inequalities with an Application
We consider a vector variational inequality in a finite-dimensional space. A new gap function is proposed, and an equivalent optimization problem for the vector variational inequality is also provided. Under some suitable conditions, we prove that the gap function is directionally differentiable and...
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| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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Wiley
2013-01-01
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| Series: | Journal of Applied Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2013/423040 |
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| _version_ | 1832552835011575808 |
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| author | Hui-qiang Ma Nan-jing Huang Meng Wu Donal O'Regan |
| author_facet | Hui-qiang Ma Nan-jing Huang Meng Wu Donal O'Regan |
| author_sort | Hui-qiang Ma |
| collection | DOAJ |
| description | We consider a vector variational inequality in a finite-dimensional space. A new gap function is proposed, and an equivalent optimization problem for the vector variational inequality is also provided. Under some suitable conditions, we prove that the gap function is directionally differentiable and that any point satisfying the first-order necessary optimality condition for the equivalent optimization problem solves the vector variational inequality. As an application, we use the new gap function to reformulate a stochastic vector variational inequality as a deterministic optimization problem. We solve this optimization problem by employing the sample average approximation method. The convergence of optimal solutions of the approximation problems is also investigated. |
| format | Article |
| id | doaj-art-2fa397e6ee0840969b94c10aa19b3e7a |
| institution | Kabale University |
| issn | 1110-757X 1687-0042 |
| language | English |
| publishDate | 2013-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Applied Mathematics |
| spelling | doaj-art-2fa397e6ee0840969b94c10aa19b3e7a2025-02-03T05:57:42ZengWileyJournal of Applied Mathematics1110-757X1687-00422013-01-01201310.1155/2013/423040423040A New Gap Function for Vector Variational Inequalities with an ApplicationHui-qiang Ma0Nan-jing Huang1Meng Wu2Donal O'Regan3School of Economics, Southwest University for Nationalities, Chengdu, Sichuan 610041, ChinaDepartment of Mathematics, Sichuan University, Chengdu, Sichuan 610064, ChinaBusiness School, Sichuan University, Chengdu, Sichuan 610064, ChinaSchool of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, IrelandWe consider a vector variational inequality in a finite-dimensional space. A new gap function is proposed, and an equivalent optimization problem for the vector variational inequality is also provided. Under some suitable conditions, we prove that the gap function is directionally differentiable and that any point satisfying the first-order necessary optimality condition for the equivalent optimization problem solves the vector variational inequality. As an application, we use the new gap function to reformulate a stochastic vector variational inequality as a deterministic optimization problem. We solve this optimization problem by employing the sample average approximation method. The convergence of optimal solutions of the approximation problems is also investigated.http://dx.doi.org/10.1155/2013/423040 |
| spellingShingle | Hui-qiang Ma Nan-jing Huang Meng Wu Donal O'Regan A New Gap Function for Vector Variational Inequalities with an Application Journal of Applied Mathematics |
| title | A New Gap Function for Vector Variational Inequalities with an Application |
| title_full | A New Gap Function for Vector Variational Inequalities with an Application |
| title_fullStr | A New Gap Function for Vector Variational Inequalities with an Application |
| title_full_unstemmed | A New Gap Function for Vector Variational Inequalities with an Application |
| title_short | A New Gap Function for Vector Variational Inequalities with an Application |
| title_sort | new gap function for vector variational inequalities with an application |
| url | http://dx.doi.org/10.1155/2013/423040 |
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