{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T13:53:50Z","timestamp":1753883630551,"version":"3.41.2"},"reference-count":11,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2011,5,31]],"date-time":"2011-05-31T00:00:00Z","timestamp":1306800000000},"content-version":"vor","delay-in-days":150,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["International Journal of Mathematics and Mathematical Sciences"],"published-print":{"date-parts":[[2011,1]]},"abstract":"<jats:p>This paper is concerned with the study of a penalization\u2010gradient algorithm for solving variational inequalities, namely, find  such that  for all <jats:italic>y<\/jats:italic> \u2208 <jats:italic>C<\/jats:italic>, where <jats:italic>A<\/jats:italic> : <jats:italic>H<\/jats:italic> \u2192 <jats:italic>H<\/jats:italic> is a single\u2010valued operator, <jats:italic>C<\/jats:italic> is a closed convex set of a real Hilbert space <jats:italic>H<\/jats:italic>. Given \u03a8 : <jats:italic>H<\/jats:italic> \u2192 <jats:italic>\u211d<\/jats:italic>\u2009\u2009 \u222a \u2009\u2009{+<jats:italic>\u221e<\/jats:italic>} which acts as a penalization function with respect to the constraint , and a penalization parameter <jats:italic>\u03b2<\/jats:italic><jats:sub><jats:italic>k<\/jats:italic><\/jats:sub>, we consider an algorithm which alternates a proximal step with respect to <jats:italic>\u2202<\/jats:italic>\u03a8 and a gradient step with respect to <jats:italic>A<\/jats:italic> and reads as <jats:italic>x<\/jats:italic><jats:sub><jats:italic>k<\/jats:italic><\/jats:sub> = (<jats:italic>I<\/jats:italic> + <jats:italic>\u03bb<\/jats:italic><jats:sub><jats:italic>k<\/jats:italic><\/jats:sub><jats:italic>\u03b2<\/jats:italic><jats:sub><jats:italic>k<\/jats:italic><\/jats:sub><jats:italic>\u2202<\/jats:italic>\u03a8)\u2009<jats:sup>\u22121<\/jats:sup>(<jats:italic>x<\/jats:italic><jats:sub><jats:italic>k<\/jats:italic>\u22121<\/jats:sub> \u2212 <jats:italic>\u03bb<\/jats:italic><jats:sub><jats:italic>k<\/jats:italic><\/jats:sub><jats:italic>A<\/jats:italic><jats:italic>x<\/jats:italic><jats:sub><jats:italic>k<\/jats:italic>\u22121<\/jats:sub>). 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\u2010point problems are also given and the multivalued case is reached by replacing the multivalued operator by its Yosida approximate which is always Lipschitz continuous.<\/jats:p>","DOI":"10.1155\/2011\/305856","type":"journal-article","created":{"date-parts":[[2011,5,31]],"date-time":"2011-05-31T19:33:32Z","timestamp":1306870412000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Penalization\u2010Gradient Algorithm for Variational Inequalities"],"prefix":"10.1155","volume":"2011","author":[{"given":"Abdellatif","family":"Moudafi","sequence":"first","affiliation":[]},{"given":"Eman","family":"Al-Shemas","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2011,5,31]]},"reference":[{"key":"e_1_2_7_1_2","first-page":"4413","article-title":"Formes bilin\u00e9aires coercitives sur les ensembles convexes","volume":"258","author":"Stampacchia G.","year":"1964","journal-title":"Comptes Rendus de l\u2032Acad\u00e9mie des Sciences de Paris"},{"key":"e_1_2_7_2_2","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1137\/100789464","article-title":"Prox-penalization and splitting methods for constrained variational problems","volume":"21","author":"Attouch H.","year":"2011","journal-title":"SIAM Journal on Optimization"},{"doi-asserted-by":"publisher","key":"e_1_2_7_3_2","DOI":"10.1016\/0022\u2010247X(79)90234\u20108"},{"key":"e_1_2_7_4_2","series-title":"International Series of Numerical Mathematics","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1007\/978-3-0348-9297-1_12","volume-title":"Trends in Mathematical Optimization","author":"Lemaire B.","year":"1988"},{"key":"e_1_2_7_5_2","series-title":"Lecture Notes in Pure and Applied Mathematics","first-page":"197","volume-title":"Optimization: Theory and Algorithms","author":"Gwinner J.","year":"1983"},{"doi-asserted-by":"publisher","key":"e_1_2_7_6_2","DOI":"10.1007\/978-3-642-02431-3"},{"unstructured":"MartinetP. 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