A Penalization-Gradient Algorithm for Variational Inequalities
This paper is concerned with the study of a penalization-gradient algorithm for solving variational inequalities, namely, find x̅∈C such that 〈Ax̅,y-x̅〉≥0 for all y∈C, where A:H→H is a single-valued operator, C is a closed convex set of a real Hilbert space H. Given Ψ:H→R ∪ {+∞} which acts as a pe...
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Wiley
2011-01-01
|
| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/2011/305856 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1850179662307655680 |
|---|---|
| author | Abdellatif Moudafi Eman Al-Shemas |
| author_facet | Abdellatif Moudafi Eman Al-Shemas |
| author_sort | Abdellatif Moudafi |
| collection | DOAJ |
| description | This paper is concerned with the study of a penalization-gradient algorithm for solving variational inequalities, namely, find x̅∈C such that 〈Ax̅,y-x̅〉≥0 for all y∈C, where A:H→H is a single-valued operator, C is a closed convex set of a real Hilbert space H. Given Ψ:H→R ∪ {+∞} which acts as a penalization function with respect to the constraint x̅∈C, and a penalization parameter βk, we consider an algorithm which alternates a proximal step with respect to ∂Ψ and a gradient step with respect to A and reads as xk=(I+λkβk∂Ψ)-1(xk-1-λkAxk-1). Under mild hypotheses, we obtain weak convergence for an inverse strongly monotone operator and strong convergence for a Lipschitz continuous and strongly monotone operator. Applications to hierarchical minimization and fixed-point problems are also given and the multivalued case is reached by replacing the multivalued operator by its Yosida approximate which is always Lipschitz continuous. |
| format | Article |
| id | doaj-art-9be57cce11cd4aa4aedbd8e0d9cdace3 |
| institution | OA Journals |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2011-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-9be57cce11cd4aa4aedbd8e0d9cdace32025-08-20T02:18:27ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252011-01-01201110.1155/2011/305856305856A Penalization-Gradient Algorithm for Variational InequalitiesAbdellatif Moudafi0Eman Al-Shemas1Département Scientifique Interfacultaires, Université des Antilles et de la Guyane, CEREGMIA, 97275 Schoelcher, Martinique, FranceDepartment of Mathematics, College of Basic Education, PAAET Main Campus-Shamiya, KuwaitThis paper is concerned with the study of a penalization-gradient algorithm for solving variational inequalities, namely, find x̅∈C such that 〈Ax̅,y-x̅〉≥0 for all y∈C, where A:H→H is a single-valued operator, C is a closed convex set of a real Hilbert space H. Given Ψ:H→R ∪ {+∞} which acts as a penalization function with respect to the constraint x̅∈C, and a penalization parameter βk, we consider an algorithm which alternates a proximal step with respect to ∂Ψ and a gradient step with respect to A and reads as xk=(I+λkβk∂Ψ)-1(xk-1-λkAxk-1). Under mild hypotheses, we obtain weak convergence for an inverse strongly monotone operator and strong convergence for a Lipschitz continuous and strongly monotone operator. Applications to hierarchical minimization and fixed-point problems are also given and the multivalued case is reached by replacing the multivalued operator by its Yosida approximate which is always Lipschitz continuous.http://dx.doi.org/10.1155/2011/305856 |
| spellingShingle | Abdellatif Moudafi Eman Al-Shemas A Penalization-Gradient Algorithm for Variational Inequalities International Journal of Mathematics and Mathematical Sciences |
| title | A Penalization-Gradient Algorithm for Variational Inequalities |
| title_full | A Penalization-Gradient Algorithm for Variational Inequalities |
| title_fullStr | A Penalization-Gradient Algorithm for Variational Inequalities |
| title_full_unstemmed | A Penalization-Gradient Algorithm for Variational Inequalities |
| title_short | A Penalization-Gradient Algorithm for Variational Inequalities |
| title_sort | penalization gradient algorithm for variational inequalities |
| url | http://dx.doi.org/10.1155/2011/305856 |
| work_keys_str_mv | AT abdellatifmoudafi apenalizationgradientalgorithmforvariationalinequalities AT emanalshemas apenalizationgradientalgorithmforvariationalinequalities AT abdellatifmoudafi penalizationgradientalgorithmforvariationalinequalities AT emanalshemas penalizationgradientalgorithmforvariationalinequalities |