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<datestamp>2017-12-21</datestamp>
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<title lang=it>A penalization-gradient algorithm for variational inequalities</title>
<creator>Moudafi, Abdellatif</creator>
<creator>Al-Shemas, Eman</creator>
<contributor>Centre de Recherche en Economie, Gestion, Modélisation et Informatique Appliquée (CEREGMIA) ; Université des Antilles et de la Guyane (UAG)</contributor>
<contributor>Department of Mathematics ; College of Basic Education</contributor>
<description>International audience</description>
<source>ISSN: 0161-1712</source>
<source>EISSN: 1687-0425</source>
<source>International Journal of Mathematics and Mathematical Sciences</source>
<publisher>Hindawi Publishing Corporation</publisher>
<identifier>hal-00776654</identifier>
<identifier>https://hal.univ-antilles.fr/hal-00776654</identifier>
<identifier>https://hal.univ-antilles.fr/hal-00776654/document</identifier>
<identifier>https://hal.univ-antilles.fr/hal-00776654/file/305856.pdf</identifier>
<source>https://hal.univ-antilles.fr/hal-00776654</source>
<source>International Journal of Mathematics and Mathematical Sciences, Hindawi Publishing Corporation, 2011, pp.1-12. 〈10.1155/2011/305856〉</source>
<identifier>DOI : 10.1155/2011/305856</identifier>
<relation>info:eu-repo/semantics/altIdentifier/doi/10.1155/2011/305856</relation>
<language>en</language>
<subject>[MATH.MATH-OC] Mathematics [math]/Optimization and Control [math.OC]</subject>
<type>info:eu-repo/semantics/article</type>
<type>Journal articles</type>
<description lang=en>This paper is concerned with the study of a penalization-gradient algorithmfor solving variational inequalities, namely, find x ∈ C such that Ax, y − x ≥ 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 → ∪ { ∞} which acts as a penalization function with respect to the constraint x ∈ 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 themultivalued operator by its Yosida approximatewhich is always Lipschitz continuous.</description>
<date>2011</date>
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