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A Penalization-Gradient Algorithm for Variational Inequalities.
- Source :
-
International Journal of Mathematics & Mathematical Sciences . 2011, p1-12. 12p. - Publication Year :
- 2011
-
Abstract
- 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. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 01611712
- Database :
- Academic Search Index
- Journal :
- International Journal of Mathematics & Mathematical Sciences
- Publication Type :
- Academic Journal
- Accession number :
- 71101043
- Full Text :
- https://doi.org/10.1155/2011/305856