{"id":"https://openalex.org/W2018632114","doi":"https://doi.org/10.1080/10556780903406589","title":"Convergence analysis of an extended auxiliary problem principle with various stopping criteria","display_name":"Convergence analysis of an extended auxiliary problem principle with various stopping criteria","publication_year":2009,"publication_date":"2009-11-16","ids":{"openalex":"https://openalex.org/W2018632114","doi":"https://doi.org/10.1080/10556780903406589","mag":"2018632114"},"language":"en","primary_location":{"id":"doi:10.1080/10556780903406589","is_oa":false,"landing_page_url":"https://doi.org/10.1080/10556780903406589","pdf_url":null,"source":{"id":"https://openalex.org/S103047102","display_name":"Optimization methods & software","issn_l":"1026-7670","issn":["1026-7670","1029-4937","1055-6788"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320547","host_organization_name":"Taylor & Francis","host_organization_lineage":["https://openalex.org/P4310320547"],"host_organization_lineage_names":["Taylor & Francis"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Optimization Methods and Software","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5034084307","display_name":"Nils Langenberg","orcid":null},"institutions":[{"id":"https://openalex.org/I89864525","display_name":"Universit\u00e4t Trier","ror":"https://ror.org/02778hg05","country_code":"DE","type":"education","lineage":["https://openalex.org/I89864525"]}],"countries":["DE"],"is_corresponding":true,"raw_author_name":"Nils Langenberg","raw_affiliation_strings":["University of Trier","Department of Mathematics, University of Trier, Trier, Germany"],"affiliations":[{"raw_affiliation_string":"University of Trier","institution_ids":["https://openalex.org/I89864525"]},{"raw_affiliation_string":"Department of Mathematics, University of Trier, Trier, Germany","institution_ids":["https://openalex.org/I89864525"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5034084307"],"corresponding_institution_ids":["https://openalex.org/I89864525"],"apc_list":null,"apc_paid":null,"fwci":1.7139,"has_fulltext":false,"cited_by_count":7,"citation_normalized_percentile":{"value":0.83619025,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"26","issue":"1","first_page":"127","last_page":"154"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10545","display_name":"Optimization and Variational Analysis","score":0.9998000264167786,"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"}},"topics":[{"id":"https://openalex.org/T10545","display_name":"Optimization and Variational Analysis","score":0.9998000264167786,"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"}},{"id":"https://openalex.org/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.986299991607666,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"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/T11502","display_name":"Facility Location and Emergency Management","score":0.9781000018119812,"subfield":{"id":"https://openalex.org/subfields/1407","display_name":"Organizational Behavior and Human Resource Management"},"field":{"id":"https://openalex.org/fields/14","display_name":"Business, Management and Accounting"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7869298458099365},{"id":"https://openalex.org/keywords/bregman-divergence","display_name":"Bregman divergence","score":0.7522730231285095},{"id":"https://openalex.org/keywords/operator","display_name":"Operator (biology)","score":0.6850733757019043},{"id":"https://openalex.org/keywords/monotone-polygon","display_name":"Monotone polygon","score":0.6270738244056702},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.6153919696807861},{"id":"https://openalex.org/keywords/variational-inequality","display_name":"Variational inequality","score":0.5534411668777466},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5197500586509705},{"id":"https://openalex.org/keywords/type","display_name":"Type (biology)","score":0.4698594808578491},{"id":"https://openalex.org/keywords/optimal-stopping","display_name":"Optimal stopping","score":0.4529607892036438},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.3413122892379761}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7869298458099365},{"id":"https://openalex.org/C149073432","wikidata":"https://www.wikidata.org/wiki/Q4960382","display_name":"Bregman divergence","level":2,"score":0.7522730231285095},{"id":"https://openalex.org/C17020691","wikidata":"https://www.wikidata.org/wiki/Q139677","display_name":"Operator (biology)","level":5,"score":0.6850733757019043},{"id":"https://openalex.org/C2834757","wikidata":"https://www.wikidata.org/wiki/Q4925424","display_name":"Monotone polygon","level":2,"score":0.6270738244056702},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.6153919696807861},{"id":"https://openalex.org/C161999928","wikidata":"https://www.wikidata.org/wiki/Q4556320","display_name":"Variational inequality","level":2,"score":0.5534411668777466},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5197500586509705},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.4698594808578491},{"id":"https://openalex.org/C99414536","wikidata":"https://www.wikidata.org/wiki/Q7098950","display_name":"Optimal stopping","level":2,"score":0.4529607892036438},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.3413122892379761},{"id":"https://openalex.org/C104317684","wikidata":"https://www.wikidata.org/wiki/Q7187","display_name":"Gene","level":2,"score":0.0},{"id":"https://openalex.org/C86339819","wikidata":"https://www.wikidata.org/wiki/Q407384","display_name":"Transcription factor","level":3,"score":0.0},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C18903297","wikidata":"https://www.wikidata.org/wiki/Q7150","display_name":"Ecology","level":1,"score":0.0},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0},{"id":"https://openalex.org/C55493867","wikidata":"https://www.wikidata.org/wiki/Q7094","display_name":"Biochemistry","level":1,"score":0.0},{"id":"https://openalex.org/C50522688","wikidata":"https://www.wikidata.org/wiki/Q189833","display_name":"Economic growth","level":1,"score":0.0},{"id":"https://openalex.org/C158448853","wikidata":"https://www.wikidata.org/wiki/Q425218","display_name":"Repressor","level":4,"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":1,"locations":[{"id":"doi:10.1080/10556780903406589","is_oa":false,"landing_page_url":"https://doi.org/10.1080/10556780903406589","pdf_url":null,"source":{"id":"https://openalex.org/S103047102","display_name":"Optimization methods & software","issn_l":"1026-7670","issn":["1026-7670","1029-4937","1055-6788"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320547","host_organization_name":"Taylor & Francis","host_organization_lineage":["https://openalex.org/P4310320547"],"host_organization_lineage_names":["Taylor & Francis"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Optimization Methods and Software","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":34,"referenced_works":["https://openalex.org/W98530473","https://openalex.org/W128756787","https://openalex.org/W163258351","https://openalex.org/W240703482","https://openalex.org/W257907672","https://openalex.org/W1495915592","https://openalex.org/W1503476534","https://openalex.org/W1532213102","https://openalex.org/W1538623501","https://openalex.org/W1549123916","https://openalex.org/W1568288633","https://openalex.org/W1966675756","https://openalex.org/W1982011875","https://openalex.org/W2007138582","https://openalex.org/W2008672897","https://openalex.org/W2019569173","https://openalex.org/W2019874425","https://openalex.org/W2021075623","https://openalex.org/W2021784794","https://openalex.org/W2023131441","https://openalex.org/W2031338222","https://openalex.org/W2038497950","https://openalex.org/W2039006402","https://openalex.org/W2047620357","https://openalex.org/W2060122892","https://openalex.org/W2065279254","https://openalex.org/W2082334903","https://openalex.org/W2090376964","https://openalex.org/W2112980792","https://openalex.org/W2150867680","https://openalex.org/W2164224215","https://openalex.org/W2170061060","https://openalex.org/W2324664502","https://openalex.org/W2747940975"],"related_works":["https://openalex.org/W2790371392","https://openalex.org/W2140505738","https://openalex.org/W4200078764","https://openalex.org/W2085771669","https://openalex.org/W2951022852","https://openalex.org/W2081685205","https://openalex.org/W2383497899","https://openalex.org/W4290851933","https://openalex.org/W2503393027","https://openalex.org/W4387225413"],"abstract_inverted_index":{"Abstract":[0],"We":[1],"investigate":[2],"the":[3,55,60,83,94,100,103,107,113,126,142,168,198,222,229,240,244,249,256,260,276],"Extended":[4,15],"Proximal":[5],"Auxiliary":[6,16],"Problem":[7,17],"Principle":[8,18],"(EPAPP)":[9],"[A.":[10],"Kaplan":[11,28,180],"and":[12,29,34,66,79,154,163,181,290,313,316,344,387],"R.":[13,30],"Tichatschke,":[14,31],"using":[19,75],"Bregman":[20,32,76,171],"distances,":[21,77],"Optimization":[22],"53":[23],"(2004),":[24],"pp.":[25,44,178],"603\u2013623,":[26],"A.":[27],"functions":[33,253],"auxiliary":[35,200,261,277],"problem":[36,288],"principle,":[37],"Optim.":[38],"Methods":[39],"Softw.":[40],"23,":[41],"1":[42],"(2008),":[43],"95\u2013107]":[45],"for":[46,72,310,354],"solving":[47],"variational":[48,286],"inequalities":[49],"whose":[50],"operator":[51,57,85,128],"\u2013":[52,58,270,275],"here":[53],"called":[54],"main":[56,84,91,127,245],"is":[59,86,136,225,266,304,320,329,340,360,373,383,389],"sum":[61],"of":[62,93,102,109,149,170,187,197,228,239,243,251,259,365],"a":[63,67,146,184,206,272,384],"maximal":[64,291],"monotone":[65,135,292],"continuous":[68],"operator.":[69,246],"As":[70],"usual":[71],"proximal":[73,160],"methods":[74],"Kalpan":[78],"Tichatschke":[80,182],"required":[81],"that":[82,125,269,358,371,394],"paramonotone.":[87],"There":[88],"are":[89],"two":[90,307],"purposes":[92],"present":[95],"article.":[96],"First,":[97],"we":[98,140,234],"establish":[99],"convergence":[101],"EPAPP":[104,143],"method":[105,144],"without":[106],"use":[108,250],"paramonotonicity":[110,223],"by":[111,116,152,322],"replacing":[112],"latter":[114],"assumption":[115],"other":[117],"(partly":[118],"rather":[119],"natural)":[120],"ones.":[121,284],"An":[122],"example":[123],"showing":[124],"does":[129],"not":[130,210],"even":[131],"have":[132],"to":[133,192,213,248,254,306],"be":[134,280,369,378],"also":[137,218],"given.":[138],"Second,":[139],"discuss":[141],"with":[145,271,391],"stopping":[147,185,207,230],"criterion":[148,186,208,231,295],"fixed-relative-error":[150],"type":[151,189],"Solodov":[153],"Svaiter":[155],"[An":[156],"inexact":[157,195],"hybrid":[158],"generalized":[159],"point":[161],"algorithm":[162],"some":[164,390,392],"new":[165],"results":[166],"on":[167],"theory":[169],"functions,":[172],"Math.":[173],"Oper.":[174],"Res.":[175],"25":[176],"(2000),":[177],"214\u2013230].":[179],"used":[183,191],"summable-error":[188],"was":[190],"admit":[193,235],"an":[194,214,236,263],"solution":[196],"corresponding":[199],"problems.":[201],"To":[202],"our":[203],"knowledge,":[204],"such":[205,393],"has":[209],"been":[211],"applied":[212],"extended":[215],"APP":[216],"yet;":[217],"in":[219,362],"this":[220,359],"situation,":[221],"hypothesis":[224],"avoided.":[226],"Independent":[227],"under":[232],"consideration,":[233],"outer":[237],"approximation":[238],"set-valued":[241],"component":[242],"Owing":[247],"Bregman-like":[252],"construct":[255],"symmetric":[257],"components":[258],"operators,":[262],"interior-point":[264],"effect":[265],"provided,":[267],"meaning":[268],"certain":[273],"precaution":[274],"problems":[278],"can":[279,377],"treated":[281],"as":[282,380],"unconstrained":[283],"Keywords:":[285],"inequalitiesauxiliary":[287],"principlemulti-valued":[289],"operatorsBregman":[293],"distancesstopping":[294],"AMS":[296],"Subject":[297],"Classification":[298],":":[299],"47H0547J2065J2065K1090C25":[300],"Acknowledgements":[301],"The":[302,327,333],"author":[303],"grateful":[305],"anonymous":[308],"referees":[309],"their":[311],"valuable":[312],"helpful":[314],"comments":[315],"remarks.":[317],"This":[318,375],"research":[319],"supported":[321],"Landesgraduiertenf\u00f6rderung":[323],"Rheinland-Pfalz.":[324],"Notes":[325],".":[326],"relation":[328],"always":[330,335],"true":[331,337],"Citation5.":[332],"equality":[334],"holds":[336],"if":[338],"f":[339],"convex,":[341],"positive":[342],"homogeneous":[343],"lsc":[345],"[Citation18,":[346],"Lemma":[347],"A.2];":[348],"nevertheless,":[349],"quite":[350],"simple":[351],"examples,":[352],"as,":[353],"example,":[355],",":[356,395],"show":[357],"false":[361],"general.":[363],"Instead":[364],"(A.7),":[366],"it":[367],"might":[368],"supposed":[370],"\u2131":[372],"monotone.":[374],"proposition":[376],"stated":[379],"follows.":[381],"If":[382],"paramonotone":[385],"operator,":[386],"there":[388],"then":[396]},"counts_by_year":[{"year":2024,"cited_by_count":1},{"year":2014,"cited_by_count":1},{"year":2012,"cited_by_count":1}],"updated_date":"2026-02-13T13:36:01.753593","created_date":"2025-10-10T00:00:00"}