{"id":"https://openalex.org/W3048977027","doi":"https://doi.org/10.1007/s11075-021-01184-9","title":"Quantitative analysis of a subgradient-type method for equilibrium problems","display_name":"Quantitative analysis of a subgradient-type method for equilibrium problems","publication_year":2021,"publication_date":"2021-08-23","ids":{"openalex":"https://openalex.org/W3048977027","doi":"https://doi.org/10.1007/s11075-021-01184-9","mag":"3048977027"},"language":"en","primary_location":{"id":"doi:10.1007/s11075-021-01184-9","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s11075-021-01184-9","pdf_url":"https://link.springer.com/content/pdf/10.1007/s11075-021-01184-9.pdf","source":{"id":"https://openalex.org/S117494069","display_name":"Numerical Algorithms","issn_l":"1017-1398","issn":["1017-1398","1572-9265"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Numerical Algorithms","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s11075-021-01184-9.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5091323905","display_name":"Nicholas Pischke","orcid":"https://orcid.org/0000-0003-1243-6787"},"institutions":[{"id":"https://openalex.org/I31512782","display_name":"Technische Universit\u00e4t Darmstadt","ror":"https://ror.org/05n911h24","country_code":"DE","type":"education","lineage":["https://openalex.org/I31512782"]}],"countries":["DE"],"is_corresponding":true,"raw_author_name":"Nicholas Pischke","raw_affiliation_strings":["Department of Mathematics, Technische Universit\u00e4t Darmstadt, Darmstadt, Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Technische Universit\u00e4t Darmstadt, Darmstadt, Germany","institution_ids":["https://openalex.org/I31512782"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5003647373","display_name":"Ulrich Kohlenbach","orcid":"https://orcid.org/0000-0002-4925-3506"},"institutions":[{"id":"https://openalex.org/I31512782","display_name":"Technische Universit\u00e4t Darmstadt","ror":"https://ror.org/05n911h24","country_code":"DE","type":"education","lineage":["https://openalex.org/I31512782"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Ulrich Kohlenbach","raw_affiliation_strings":["Department of Mathematics, Technische Universit\u00e4t Darmstadt, Darmstadt, Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Technische Universit\u00e4t Darmstadt, Darmstadt, Germany","institution_ids":["https://openalex.org/I31512782"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5091323905"],"corresponding_institution_ids":["https://openalex.org/I31512782"],"apc_list":{"value":2390,"currency":"EUR","value_usd":2990},"apc_paid":{"value":2390,"currency":"EUR","value_usd":2990},"fwci":0.4566,"has_fulltext":true,"cited_by_count":3,"citation_normalized_percentile":{"value":0.65738668,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"90","issue":"1","first_page":"197","last_page":"219"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10545","display_name":"Optimization and Variational Analysis","score":0.9998999834060669,"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":0.9998999834060669,"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":0.9984999895095825,"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/T11777","display_name":"Fixed Point Theorems Analysis","score":0.9829000234603882,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/subgradient-method","display_name":"Subgradient method","score":0.9183196425437927},{"id":"https://openalex.org/keywords/theory-of-computation","display_name":"Theory of computation","score":0.7361583709716797},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7222659587860107},{"id":"https://openalex.org/keywords/monotonic-function","display_name":"Monotonic function","score":0.6558140516281128},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.5697198510169983},{"id":"https://openalex.org/keywords/type","display_name":"Type (biology)","score":0.5312609076499939},{"id":"https://openalex.org/keywords/metric","display_name":"Metric (unit)","score":0.4759359359741211},{"id":"https://openalex.org/keywords/regular-polygon","display_name":"Regular polygon","score":0.4417800307273865},{"id":"https://openalex.org/keywords/mathematical-economics","display_name":"Mathematical economics","score":0.4272249639034271},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.4145634174346924},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.3743063509464264},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.2706409692764282},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.10922938585281372},{"id":"https://openalex.org/keywords/economics","display_name":"Economics","score":0.07130485773086548}],"concepts":[{"id":"https://openalex.org/C158968445","wikidata":"https://www.wikidata.org/wiki/Q7631150","display_name":"Subgradient method","level":2,"score":0.9183196425437927},{"id":"https://openalex.org/C24858836","wikidata":"https://www.wikidata.org/wiki/Q844718","display_name":"Theory of computation","level":2,"score":0.7361583709716797},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7222659587860107},{"id":"https://openalex.org/C72169020","wikidata":"https://www.wikidata.org/wiki/Q194404","display_name":"Monotonic function","level":2,"score":0.6558140516281128},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.5697198510169983},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.5312609076499939},{"id":"https://openalex.org/C176217482","wikidata":"https://www.wikidata.org/wiki/Q860554","display_name":"Metric (unit)","level":2,"score":0.4759359359741211},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.4417800307273865},{"id":"https://openalex.org/C144237770","wikidata":"https://www.wikidata.org/wiki/Q747534","display_name":"Mathematical economics","level":1,"score":0.4272249639034271},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.4145634174346924},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3743063509464264},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.2706409692764282},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.10922938585281372},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.07130485773086548},{"id":"https://openalex.org/C18903297","wikidata":"https://www.wikidata.org/wiki/Q7150","display_name":"Ecology","level":1,"score":0.0},{"id":"https://openalex.org/C21547014","wikidata":"https://www.wikidata.org/wiki/Q1423657","display_name":"Operations management","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/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","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:10.1007/s11075-021-01184-9","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s11075-021-01184-9","pdf_url":"https://link.springer.com/content/pdf/10.1007/s11075-021-01184-9.pdf","source":{"id":"https://openalex.org/S117494069","display_name":"Numerical Algorithms","issn_l":"1017-1398","issn":["1017-1398","1572-9265"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Numerical Algorithms","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:2008.06900","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2008.06900","pdf_url":"https://arxiv.org/pdf/2008.06900","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"pmh:oai:tubiblio.ulb.tu-darmstadt.de:150003","is_oa":false,"landing_page_url":"http://tubiblio.ulb.tu-darmstadt.de/150003/","pdf_url":null,"source":{"id":"https://openalex.org/S4377196390","display_name":"TUbilio (Technical University of Darmstadt)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I31512782","host_organization_name":"Technische Universit\u00e4t Darmstadt","host_organization_lineage":["https://openalex.org/I31512782"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Artikel"}],"best_oa_location":{"id":"doi:10.1007/s11075-021-01184-9","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s11075-021-01184-9","pdf_url":"https://link.springer.com/content/pdf/10.1007/s11075-021-01184-9.pdf","source":{"id":"https://openalex.org/S117494069","display_name":"Numerical Algorithms","issn_l":"1017-1398","issn":["1017-1398","1572-9265"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Numerical Algorithms","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G7137432809","display_name":null,"funder_award_id":"DFG KO 1737/6-1","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"},{"id":"https://openalex.org/G7776082359","display_name":null,"funder_award_id":"1737/6-1","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"},{"id":"https://openalex.org/G911389773","display_name":null,"funder_award_id":"KO 1737/6-1","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"}],"funders":[{"id":"https://openalex.org/F4320320879","display_name":"Deutsche Forschungsgemeinschaft","ror":"https://ror.org/018mejw64"},{"id":"https://openalex.org/F4320323384","display_name":"Technische Universit\u00e4t Darmstadt","ror":"https://ror.org/05n911h24"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3048977027.pdf","grobid_xml":"https://content.openalex.org/works/W3048977027.grobid-xml"},"referenced_works_count":24,"referenced_works":["https://openalex.org/W205960364","https://openalex.org/W561202933","https://openalex.org/W1545819195","https://openalex.org/W1546168022","https://openalex.org/W1853237406","https://openalex.org/W2007269342","https://openalex.org/W2025454433","https://openalex.org/W2067914884","https://openalex.org/W2068357778","https://openalex.org/W2077607486","https://openalex.org/W2278644336","https://openalex.org/W2611511037","https://openalex.org/W2962728982","https://openalex.org/W2962941654","https://openalex.org/W2963098769","https://openalex.org/W2963798168","https://openalex.org/W2972486801","https://openalex.org/W3048977027","https://openalex.org/W3099600669","https://openalex.org/W3103639684","https://openalex.org/W4229650096","https://openalex.org/W4243563333","https://openalex.org/W4254573138","https://openalex.org/W7052723095"],"related_works":["https://openalex.org/W2359120930","https://openalex.org/W2037613239","https://openalex.org/W2781777036","https://openalex.org/W2390469630","https://openalex.org/W2059050877","https://openalex.org/W1971290991","https://openalex.org/W1944783162","https://openalex.org/W1984031944","https://openalex.org/W1990626142","https://openalex.org/W4238880966"],"abstract_inverted_index":{"Abstract":[0],"We":[1],"use":[2],"techniques":[3],"originating":[4],"from":[5],"the":[6,31,68,71,85],"subdiscipline":[7],"of":[8,17,24,40,61,70,76,79],"mathematical":[9],"logic":[10],"called":[11],"\u2018proof":[12],"mining\u2019":[13],"to":[14,48],"provide":[15],"rates":[16],"metastability":[18],"and\u2014under":[19],"a":[20,27,58],"metric":[21],"regularity":[22],"assumption\u2014rates":[23],"convergence":[25],"for":[26],"subgradient-type":[28],"algorithm":[29,45],"solving":[30],"equilibrium":[32],"problem":[33],"in":[34,55,88],"convex":[35],"optimization":[36],"over":[37],"fixed-point":[38],"sets":[39],"firmly":[41],"nonexpansive":[42],"mappings.":[43],"The":[44],"is":[46],"due":[47],"H.":[49],"Iiduka":[50],"and":[51],"I.":[52],"Yamada":[53],"who":[54],"2009":[56],"gave":[57],"noneffective":[59],"proof":[60],"its":[62],"convergence.":[63],"This":[64],"case":[65],"study":[66],"illustrates":[67],"applicability":[69],"logic-based":[72],"abstract":[73],"quantitative":[74],"analysis":[75],"general":[77],"forms":[78],"Fej\u00e9r":[80],"monotonicity":[81],"as":[82],"given":[83],"by":[84],"second":[86],"author":[87],"previous":[89],"papers.":[90]},"counts_by_year":[{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":1},{"year":2021,"cited_by_count":1}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}
