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A Penalization-Gradient Algorithm for Variational Inequalities
Joint Authors
al-Shemas, Eman
Moudafi, Abdellatif
Source
International Journal of Mathematics and Mathematical Sciences
Issue
Vol. 2011, Issue 2011 (31 Dec. 2011), pp.1-12, 12 p.
Publisher
Hindawi Publishing Corporation
Publication Date
2011-05-31
Country of Publication
Egypt
No. of Pages
12
Main Subjects
Abstract EN
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.
American Psychological Association (APA)
Moudafi, Abdellatif& al-Shemas, Eman. 2011. A Penalization-Gradient Algorithm for Variational Inequalities. International Journal of Mathematics and Mathematical Sciences،Vol. 2011, no. 2011, pp.1-12.
https://search.emarefa.net/detail/BIM-461989
Modern Language Association (MLA)
Moudafi, Abdellatif& al-Shemas, Eman. A Penalization-Gradient Algorithm for Variational Inequalities. International Journal of Mathematics and Mathematical Sciences No. 2011 (2011), pp.1-12.
https://search.emarefa.net/detail/BIM-461989
American Medical Association (AMA)
Moudafi, Abdellatif& al-Shemas, Eman. A Penalization-Gradient Algorithm for Variational Inequalities. International Journal of Mathematics and Mathematical Sciences. 2011. Vol. 2011, no. 2011, pp.1-12.
https://search.emarefa.net/detail/BIM-461989
Data Type
Journal Articles
Language
English
Notes
Includes bibliographical references
Record ID
BIM-461989