{"id":"https://openalex.org/W2077726563","doi":"https://doi.org/10.1137/110820300","title":"Coupling Forward-Backward with Penalty Schemes and Parallel Splitting for Constrained Variational Inequalities","display_name":"Coupling Forward-Backward with Penalty Schemes and Parallel Splitting for Constrained Variational Inequalities","publication_year":2011,"publication_date":"2011-10-01","ids":{"openalex":"https://openalex.org/W2077726563","doi":"https://doi.org/10.1137/110820300","mag":"2077726563"},"language":"en","primary_location":{"id":"doi:10.1137/110820300","is_oa":false,"landing_page_url":"https://doi.org/10.1137/110820300","pdf_url":null,"source":{"id":"https://openalex.org/S928796702","display_name":"SIAM Journal on Optimization","issn_l":"1052-6234","issn":["1052-6234","1095-7189"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Optimization","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://hal.science/hal-00803596","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5038818485","display_name":"H\u00e9dy Attouch","orcid":"https://orcid.org/0000-0003-2676-0887"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"H\u00e9dy Attouch","raw_affiliation_strings":["attouch@math.univ-montp2.fr and marco@math.univ-montp2.fr#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"attouch@math.univ-montp2.fr and marco@math.univ-montp2.fr#TAB#","institution_ids":[]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5071460705","display_name":"Marc-Olivier Czarnecki","orcid":null},"institutions":[{"id":"https://openalex.org/I4210142014","display_name":"Institut Montpelli\u00e9rain Alexander Grothendieck","ror":"https://ror.org/044kxby82","country_code":"FR","type":"facility","lineage":["https://openalex.org/I1294671590","https://openalex.org/I1294671590","https://openalex.org/I19894307","https://openalex.org/I4210141950","https://openalex.org/I4210142014"]}],"countries":["FR"],"is_corresponding":false,"raw_author_name":"Marc-Olivier Czarnecki","raw_affiliation_strings":["Institut de Math\u00e9matiques et de Mod\u00e9lisation de Montpellier"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institut de Math\u00e9matiques et de Mod\u00e9lisation de Montpellier","institution_ids":["https://openalex.org/I4210142014"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5053427654","display_name":"Juan Peypouquet","orcid":"https://orcid.org/0000-0002-8551-0522"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Juan Peypouquet","raw_affiliation_strings":["juan.peypouquet@usm.cl#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"juan.peypouquet@usm.cl#TAB#","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":4.0994,"has_fulltext":false,"cited_by_count":66,"citation_normalized_percentile":{"value":0.94100967,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"21","issue":"4","first_page":"1251","last_page":"1274"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.9991000294685364,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.9991000294685364,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11205","display_name":"Numerical methods in inverse problems","score":0.9954000115394592,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10545","display_name":"Optimization and Variational Analysis","score":0.9951000213623047,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.798734188079834},{"id":"https://openalex.org/keywords/nabla-symbol","display_name":"Nabla symbol","score":0.7576189041137695},{"id":"https://openalex.org/keywords/lipschitz-continuity","display_name":"Lipschitz continuity","score":0.7210640907287598},{"id":"https://openalex.org/keywords/variational-inequality","display_name":"Variational inequality","score":0.6524471044540405},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.6184384226799011},{"id":"https://openalex.org/keywords/differentiable-function","display_name":"Differentiable function","score":0.5107364654541016},{"id":"https://openalex.org/keywords/monotone-polygon","display_name":"Monotone polygon","score":0.48792168498039246},{"id":"https://openalex.org/keywords/sequence","display_name":"Sequence (biology)","score":0.47965410351753235},{"id":"https://openalex.org/keywords/convex-function","display_name":"Convex function","score":0.4739651381969452},{"id":"https://openalex.org/keywords/subderivative","display_name":"Subderivative","score":0.42210376262664795},{"id":"https://openalex.org/keywords/regular-polygon","display_name":"Regular polygon","score":0.3772713243961334},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.3690330386161804},{"id":"https://openalex.org/keywords/convex-optimization","display_name":"Convex optimization","score":0.23111730813980103},{"id":"https://openalex.org/keywords/omega","display_name":"Omega","score":0.22471603751182556},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.09956437349319458},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.09776130318641663}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.798734188079834},{"id":"https://openalex.org/C54207081","wikidata":"https://www.wikidata.org/wiki/Q937750","display_name":"Nabla symbol","level":3,"score":0.7576189041137695},{"id":"https://openalex.org/C22324862","wikidata":"https://www.wikidata.org/wiki/Q652707","display_name":"Lipschitz continuity","level":2,"score":0.7210640907287598},{"id":"https://openalex.org/C161999928","wikidata":"https://www.wikidata.org/wiki/Q4556320","display_name":"Variational inequality","level":2,"score":0.6524471044540405},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6184384226799011},{"id":"https://openalex.org/C202615002","wikidata":"https://www.wikidata.org/wiki/Q783507","display_name":"Differentiable function","level":2,"score":0.5107364654541016},{"id":"https://openalex.org/C2834757","wikidata":"https://www.wikidata.org/wiki/Q4925424","display_name":"Monotone polygon","level":2,"score":0.48792168498039246},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.47965410351753235},{"id":"https://openalex.org/C145446738","wikidata":"https://www.wikidata.org/wiki/Q319913","display_name":"Convex function","level":3,"score":0.4739651381969452},{"id":"https://openalex.org/C200661725","wikidata":"https://www.wikidata.org/wiki/Q289500","display_name":"Subderivative","level":4,"score":0.42210376262664795},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.3772713243961334},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3690330386161804},{"id":"https://openalex.org/C157972887","wikidata":"https://www.wikidata.org/wiki/Q463359","display_name":"Convex optimization","level":3,"score":0.23111730813980103},{"id":"https://openalex.org/C2779557605","wikidata":"https://www.wikidata.org/wiki/Q9890","display_name":"Omega","level":2,"score":0.22471603751182556},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.09956437349319458},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.09776130318641663},{"id":"https://openalex.org/C54355233","wikidata":"https://www.wikidata.org/wiki/Q7162","display_name":"Genetics","level":1,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1137/110820300","is_oa":false,"landing_page_url":"https://doi.org/10.1137/110820300","pdf_url":null,"source":{"id":"https://openalex.org/S928796702","display_name":"SIAM Journal on Optimization","issn_l":"1052-6234","issn":["1052-6234","1095-7189"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Optimization","raw_type":"journal-article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.739.5585","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.739.5585","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://hal.inria.fr/docs/00/80/35/96/PDF/att-cza-pey-siopt-2011-2.pdf","raw_type":"text"},{"id":"pmh:oai:HAL:hal-00803596v1","is_oa":true,"landing_page_url":"https://hal.science/hal-00803596","pdf_url":null,"source":{"id":"https://openalex.org/S4306402512","display_name":"HAL (Le Centre pour la Communication Scientifique Directe)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1294671590","host_organization_name":"Centre National de la Recherche Scientifique","host_organization_lineage":["https://openalex.org/I1294671590"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"SIAM Journal on Optimization, 2011, 21 (4), pp.1251-1274. &#x27E8;10.1137/110820300&#x27E9;","raw_type":"info:eu-repo/semantics/article"}],"best_oa_location":{"id":"pmh:oai:HAL:hal-00803596v1","is_oa":true,"landing_page_url":"https://hal.science/hal-00803596","pdf_url":null,"source":{"id":"https://openalex.org/S4306402512","display_name":"HAL (Le Centre pour la Communication Scientifique Directe)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1294671590","host_organization_name":"Centre National de la Recherche Scientifique","host_organization_lineage":["https://openalex.org/I1294671590"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"SIAM Journal on Optimization, 2011, 21 (4), pp.1251-1274. &#x27E8;10.1137/110820300&#x27E9;","raw_type":"info:eu-repo/semantics/article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G2928036526","display_name":"Optimisation, Strat\u00e9gies, Selection : Dynamiques, Approximations, et Applications","funder_award_id":"ANR-08-BLAN-0294","funder_id":"https://openalex.org/F4320320883","funder_display_name":"Agence Nationale de la Recherche"}],"funders":[{"id":"https://openalex.org/F4320320883","display_name":"Agence Nationale de la Recherche","ror":"https://ror.org/00rbzpz17"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":34,"referenced_works":["https://openalex.org/W1021395667","https://openalex.org/W1572548753","https://openalex.org/W1582617215","https://openalex.org/W1969666805","https://openalex.org/W1979854129","https://openalex.org/W1986931325","https://openalex.org/W1996902173","https://openalex.org/W1998576973","https://openalex.org/W2001702564","https://openalex.org/W2005009439","https://openalex.org/W2005833769","https://openalex.org/W2006262045","https://openalex.org/W2011363942","https://openalex.org/W2012117745","https://openalex.org/W2013921128","https://openalex.org/W2019613464","https://openalex.org/W2045556880","https://openalex.org/W2047515743","https://openalex.org/W2053487511","https://openalex.org/W2092554297","https://openalex.org/W2095030074","https://openalex.org/W2135046866","https://openalex.org/W2145096794","https://openalex.org/W2157044855","https://openalex.org/W2163916252","https://openalex.org/W2164452299","https://openalex.org/W2296616510","https://openalex.org/W2315925303","https://openalex.org/W2569816327","https://openalex.org/W2963705414","https://openalex.org/W3127840685","https://openalex.org/W3211220023","https://openalex.org/W4250955649","https://openalex.org/W4285719527"],"related_works":["https://openalex.org/W2229925025","https://openalex.org/W1539596337","https://openalex.org/W33469805","https://openalex.org/W1855007440","https://openalex.org/W2788247819","https://openalex.org/W2072255091","https://openalex.org/W2761731838","https://openalex.org/W4221091651","https://openalex.org/W2084754442","https://openalex.org/W2356958610"],"abstract_inverted_index":{"We":[0,146,214],"are":[1],"concerned":[2],"with":[3,73],"the":[4,40,76,102,114,121,133,152,159,176,179,212],"study":[5],"of":[6,9,84,94,124,132,151,161,171,178,201,229],"a":[7,25,33,45,57,70,81,92,140,162,168],"class":[8],"forward-backward":[10,104],"penalty":[11],"schemes":[12],"for":[13,139,206,223],"solving":[14],"variational":[15,207],"inequalities":[16,208],"$0\\in":[17],"Ax":[18],"+":[19],"N_C":[20],"(x)$":[21],"where":[22],"$\\mathcal{H}$":[23],"is":[24,32,39,63,120,158,232],"real":[26],"Hilbert":[27],"space,":[28],"$A:":[29],"\\mathcal{H}\\rightrightarrows":[30],"\\mathcal{H}$":[31],"maximal":[34,142],"monotone":[35,143],"operator,":[36],"and":[37,66,91,197,218],"$N_C$":[38],"outward":[41],"normal":[42],"cone":[43],"to":[44,75,89,136,187,194],"closed":[46],"convex":[47,58,165],"set":[48],"$C\\subset\\mathcal{H}$.":[49],"Let":[50],"$\\Psi:":[51],"\\mathcal{H}":[52],"\\to":[53],"\\mathbb":[54],"R$":[55],"be":[56],"differentiable":[59],"function":[60,72],"whose":[61],"gradient":[62],"Lipschitz":[64,122],"continuous":[65],"which":[67,87],"acts":[68],"as":[69],"penalization":[71,85],"respect":[74],"constraint":[77],"$x\\in":[78],"C.$":[79],"Given":[80],"sequence":[82,93,134,154],"$(\\beta_n)$":[83,112],"parameters":[86],"tends":[88],"infinity,":[90],"positive":[95],"time":[96],"steps":[97],"$(\\lambda_n)":[98],"\\in\\ell^2\\setminus\\ell^1$,":[99],"we":[100,127,174,190],"consider":[101],"diagonal":[103],"algorithm":[105],"$x_{n+1}=(I+\\lambda_nA)^{-1}(x_n-\\lambda_n\\beta_n":[106],"\\nabla":[107],"\\Psi":[108],"(x_n)).$":[109],"Assuming":[110],"that":[111,221],"satisfies":[113],"growth":[115],"condition":[116],"$\\limsup_{n\\to\\infty}\\lambda_n\\beta_n<2/\\theta$":[117],"(where":[118],"$\\theta$":[119],"constant":[123],"$\\nabla":[125,181],"\\Psi$),":[126],"obtain":[128,148,198],"weak":[129,149],"ergodic":[130],"convergence":[131,150,199],"$(x_n)$":[135,155],"an":[137],"equilibrium":[138],"general":[141],"operator":[144,180],"A.":[145],"also":[147,215],"whole":[153],"when":[156],"A":[157],"subdifferential":[160],"proper":[163],"lower-semicontinuous":[164],"function.":[166],"As":[167],"key":[169],"ingredient":[170],"our":[172,185],"analysis,":[173],"use":[175],"cocoerciveness":[177],"\\Psi$.":[182],"When":[183],"specializing":[184],"results":[186,200,220],"coupled":[188],"systems,":[189],"bring":[191],"new":[192,202],"light":[193],"Passty's":[195],"theorem":[196],"parallel":[203],"splitting":[204],"algorithms":[205],"involving":[209],"coupling":[210],"in":[211],"constraint.":[213],"establish":[216],"robustness":[217],"stability":[219],"account":[222],"numerical":[224],"approximation":[225],"errors.":[226],"An":[227],"illustration":[228],"compressive":[230],"sensing":[231],"given.":[233]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":3},{"year":2024,"cited_by_count":4},{"year":2023,"cited_by_count":3},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":5},{"year":2020,"cited_by_count":4},{"year":2019,"cited_by_count":4},{"year":2018,"cited_by_count":8},{"year":2017,"cited_by_count":6},{"year":2016,"cited_by_count":8},{"year":2015,"cited_by_count":7},{"year":2014,"cited_by_count":7},{"year":2013,"cited_by_count":3}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}