{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,22]],"date-time":"2026-08-22T09:21:05Z","timestamp":1787390465066,"version":"build-2736575974"},"reference-count":22,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Optim."],"published-print":{"date-parts":[[1998,2]]},"abstract":"<jats:p>We consider a generalized proximal point method for solving variational inequality problems with monotone operators in a Hilbert space. It differs from the classical proximal point method (as discussed by Rockafellar for the problem of finding zeroes of monotone operators) in the use of generalized distances, called Bregman distances, instead of the Euclidean one. These distances play not only a regularization role but also a penalization one, forcing the sequence generated by the method to remain in the interior of the feasible set so that the method becomes an interior point one. Under appropriate assumptions on the Bregman distance and the monotone operator we prove that the sequence converges (weakly) if and only if the problem has solutions, in which case the weak limit is a solution. If the problem does not have solutions, then the sequence is unbounded. We extend similar previous results for the proximal point method with Bregman distances which dealt only with the finite dimensional case and which applied only to convex optimization problems or to finding zeroes of monotone operators, which are particular cases of variational inequality problems.<\/jats:p>","DOI":"10.1137\/s1052623495286302","type":"journal-article","created":{"date-parts":[[2003,6,11]],"date-time":"2003-06-11T11:12:06Z","timestamp":1055329926000},"page":"197-216","source":"Crossref","is-referenced-by-count":116,"title":["A Generalized Proximal Point Algorithm for the Variational Inequality Problem in a Hilbert Space"],"prefix":"10.1137","volume":"8","author":[{"given":"Regina S.","family":"Burachik","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Alfredo N.","family":"Iusem","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,7,31]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1016\/0041\u20105553(67)90040\u20107"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1007\/BF02756796"},{"key":"R3","unstructured":"H. Br\u00e9zis,\n                      Op\u00e9rateurs Monotones Maximaux et Semigroups de Contractions dans les Espaces de Hilbert\n                      , Mathematics Studies 5, North\u2013Holland, New York, 1973."},{"key":"R4","unstructured":"F. E. Browder,\n                      Nonlinear Operators and Nonlinear Equations of Evolution in Banach Spaces\n                      , Proc. Sympos. Pure Math. 18, American Mathematical Society, Providence, RI, 1976."},{"key":"R5","unstructured":"R. S. 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Convex Anal."},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1007\/BF00932654"},{"key":"R16","unstructured":"D. Kinderlerer, G. Stampakk\u2019ya, Vvedenie v variatsionnye neravenstva i ikh prilozheniya, \u201cMir\u201d, 1983, 256\u20130, Translated from the English by G. G. Magaril\u2010Il\u2019yaev and A. V. Fursikov; Translation edited and with a preface by V. M. Tikhomirov85b:49007"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1137\/S0363012995281742"},{"key":"R18","unstructured":"Bernard Lemaire, The proximal algorithm, Internat. Schriftenreihe Numer. Math., Vol. 87, Birkh\u00e4user, Basel, 1989, 73\u20138790f:90108"},{"key":"R19","first-page":"397","volume":"36","author":"Yuan Xian","year":"1993","journal-title":"Acta Math. Sinica"},{"key":"R20","unstructured":"R. T. Rockafellar,\n                      Convex Analysis\n                      , Princeton University Press, Princeton, NJ, 1970."},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1970-0282272-5"},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1137\/0314056"}],"container-title":["SIAM Journal on Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/S1052623495286302","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:01:49Z","timestamp":1787331709000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/S1052623495286302"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1998,2]]},"references-count":22,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1998,2]]}},"alternative-id":["10.1137\/S1052623495286302"],"URL":"https:\/\/doi.org\/10.1137\/s1052623495286302","relation":{},"ISSN":["1052-6234","1095-7189"],"issn-type":[{"value":"1052-6234","type":"print"},{"value":"1095-7189","type":"electronic"}],"subject":[],"published":{"date-parts":[[1998,2]]}}}