{"id":"https://openalex.org/W2058164831","doi":"https://doi.org/10.1007/s10589-013-9599-7","title":"On the O(1/t) convergence rate of the projection and contraction methods for variational inequalities with Lipschitz continuous monotone operators","display_name":"On the O(1/t) convergence rate of the projection and contraction methods for variational inequalities with Lipschitz continuous monotone operators","publication_year":2013,"publication_date":"2013-09-24","ids":{"openalex":"https://openalex.org/W2058164831","doi":"https://doi.org/10.1007/s10589-013-9599-7","mag":"2058164831"},"language":"en","primary_location":{"id":"doi:10.1007/s10589-013-9599-7","is_oa":false,"landing_page_url":"https://doi.org/10.1007/s10589-013-9599-7","pdf_url":null,"source":{"id":"https://openalex.org/S39243548","display_name":"Computational Optimization and Applications","issn_l":"0926-6003","issn":["0926-6003","1573-2894"],"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":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Computational Optimization and Applications","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/A5102801082","display_name":"Xingju Cai","orcid":"https://orcid.org/0000-0002-6957-477X"},"institutions":[{"id":"https://openalex.org/I881766915","display_name":"Nanjing University","ror":"https://ror.org/01rxvg760","country_code":"CN","type":"education","lineage":["https://openalex.org/I881766915"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Xingju Cai","raw_affiliation_strings":["Department of Mathematics, Nanjing University, Nanjing, 210093, P.R. China","Department of Mathematics, Nanjing University, Nanjing, P.R. China 210093"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Nanjing University, Nanjing, 210093, P.R. China","institution_ids":["https://openalex.org/I881766915"]},{"raw_affiliation_string":"Department of Mathematics, Nanjing University, Nanjing, P.R. China 210093","institution_ids":["https://openalex.org/I881766915"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5068750984","display_name":"Guoyong Gu","orcid":"https://orcid.org/0000-0003-4933-0679"},"institutions":[{"id":"https://openalex.org/I881766915","display_name":"Nanjing University","ror":"https://ror.org/01rxvg760","country_code":"CN","type":"education","lineage":["https://openalex.org/I881766915"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Guoyong Gu","raw_affiliation_strings":["Department of Mathematics, Nanjing University, Nanjing, 210093, P.R. China","Department of Mathematics, Nanjing University, Nanjing, P.R. China 210093"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Nanjing University, Nanjing, 210093, P.R. China","institution_ids":["https://openalex.org/I881766915"]},{"raw_affiliation_string":"Department of Mathematics, Nanjing University, Nanjing, P.R. China 210093","institution_ids":["https://openalex.org/I881766915"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5039946576","display_name":"Bingsheng He","orcid":"https://orcid.org/0000-0001-8618-4581"},"institutions":[{"id":"https://openalex.org/I881766915","display_name":"Nanjing University","ror":"https://ror.org/01rxvg760","country_code":"CN","type":"education","lineage":["https://openalex.org/I881766915"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Bingsheng He","raw_affiliation_strings":["Department of Mathematics, Nanjing University, Nanjing, 210093, P.R. China","Department of Mathematics, Nanjing University, Nanjing, P.R. China 210093"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Nanjing University, Nanjing, 210093, P.R. China","institution_ids":["https://openalex.org/I881766915"]},{"raw_affiliation_string":"Department of Mathematics, Nanjing University, Nanjing, P.R. China 210093","institution_ids":["https://openalex.org/I881766915"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5102801082"],"corresponding_institution_ids":["https://openalex.org/I881766915"],"apc_list":{"value":2390,"currency":"EUR","value_usd":2990},"apc_paid":null,"fwci":4.1027,"has_fulltext":false,"cited_by_count":117,"citation_normalized_percentile":{"value":0.940106,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":94,"max":100},"biblio":{"volume":"57","issue":"2","first_page":"339","last_page":"363"},"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":0.9936000108718872,"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":0.9897000193595886,"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/variational-inequality","display_name":"Variational inequality","score":0.8286501169204712},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8202094435691833},{"id":"https://openalex.org/keywords/lipschitz-continuity","display_name":"Lipschitz continuity","score":0.7898350954055786},{"id":"https://openalex.org/keywords/monotone-polygon","display_name":"Monotone polygon","score":0.7696828246116638},{"id":"https://openalex.org/keywords/contraction","display_name":"Contraction (grammar)","score":0.720396101474762},{"id":"https://openalex.org/keywords/rate-of-convergence","display_name":"Rate of convergence","score":0.6879070997238159},{"id":"https://openalex.org/keywords/monotonic-function","display_name":"Monotonic function","score":0.6461423635482788},{"id":"https://openalex.org/keywords/projection-method","display_name":"Projection method","score":0.5420215725898743},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5171656012535095},{"id":"https://openalex.org/keywords/projection","display_name":"Projection (relational algebra)","score":0.5112664699554443},{"id":"https://openalex.org/keywords/strongly-monotone","display_name":"Strongly monotone","score":0.5010757446289062},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.41190487146377563},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.28634968400001526},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.24338805675506592},{"id":"https://openalex.org/keywords/dykstras-projection-algorithm","display_name":"Dykstra's projection algorithm","score":0.21117651462554932},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.12876388430595398},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.065562903881073}],"concepts":[{"id":"https://openalex.org/C161999928","wikidata":"https://www.wikidata.org/wiki/Q4556320","display_name":"Variational inequality","level":2,"score":0.8286501169204712},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8202094435691833},{"id":"https://openalex.org/C22324862","wikidata":"https://www.wikidata.org/wiki/Q652707","display_name":"Lipschitz continuity","level":2,"score":0.7898350954055786},{"id":"https://openalex.org/C2834757","wikidata":"https://www.wikidata.org/wiki/Q4925424","display_name":"Monotone polygon","level":2,"score":0.7696828246116638},{"id":"https://openalex.org/C163415756","wikidata":"https://www.wikidata.org/wiki/Q126473","display_name":"Contraction (grammar)","level":2,"score":0.720396101474762},{"id":"https://openalex.org/C57869625","wikidata":"https://www.wikidata.org/wiki/Q1783502","display_name":"Rate of convergence","level":3,"score":0.6879070997238159},{"id":"https://openalex.org/C72169020","wikidata":"https://www.wikidata.org/wiki/Q194404","display_name":"Monotonic function","level":2,"score":0.6461423635482788},{"id":"https://openalex.org/C65557600","wikidata":"https://www.wikidata.org/wiki/Q7249451","display_name":"Projection method","level":3,"score":0.5420215725898743},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5171656012535095},{"id":"https://openalex.org/C57493831","wikidata":"https://www.wikidata.org/wiki/Q3134666","display_name":"Projection (relational algebra)","level":2,"score":0.5112664699554443},{"id":"https://openalex.org/C12715386","wikidata":"https://www.wikidata.org/wiki/Q7624691","display_name":"Strongly monotone","level":3,"score":0.5010757446289062},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.41190487146377563},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.28634968400001526},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.24338805675506592},{"id":"https://openalex.org/C202426404","wikidata":"https://www.wikidata.org/wiki/Q5318686","display_name":"Dykstra's projection algorithm","level":2,"score":0.21117651462554932},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.12876388430595398},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.065562903881073},{"id":"https://openalex.org/C71924100","wikidata":"https://www.wikidata.org/wiki/Q11190","display_name":"Medicine","level":0,"score":0.0},{"id":"https://openalex.org/C126322002","wikidata":"https://www.wikidata.org/wiki/Q11180","display_name":"Internal medicine","level":1,"score":0.0},{"id":"https://openalex.org/C127162648","wikidata":"https://www.wikidata.org/wiki/Q16858953","display_name":"Channel (broadcasting)","level":2,"score":0.0},{"id":"https://openalex.org/C31258907","wikidata":"https://www.wikidata.org/wiki/Q1301371","display_name":"Computer network","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1007/s10589-013-9599-7","is_oa":false,"landing_page_url":"https://doi.org/10.1007/s10589-013-9599-7","pdf_url":null,"source":{"id":"https://openalex.org/S39243548","display_name":"Computational Optimization and Applications","issn_l":"0926-6003","issn":["0926-6003","1573-2894"],"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":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Computational Optimization and Applications","raw_type":"journal-article"},{"id":"pmh:oai:RePEc:spr:coopap:v:57:y:2014:i:2:p:339-363","is_oa":false,"landing_page_url":"http://hdl.handle.net/10.1007/s10589-013-9599-7","pdf_url":null,"source":{"id":"https://openalex.org/S4306401271","display_name":"RePEc: Research Papers in Economics","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I77793887","host_organization_name":"Federal Reserve Bank of St. Louis","host_organization_lineage":["https://openalex.org/I77793887"],"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":"article"}],"best_oa_location":null,"sustainable_development_goals":[{"id":"https://metadata.un.org/sdg/10","score":0.5799999833106995,"display_name":"Reduced inequalities"}],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":25,"referenced_works":["https://openalex.org/W649579232","https://openalex.org/W1488903562","https://openalex.org/W1519983590","https://openalex.org/W1543439990","https://openalex.org/W1571006235","https://openalex.org/W1574923065","https://openalex.org/W1603765807","https://openalex.org/W1945940342","https://openalex.org/W1966251519","https://openalex.org/W1979735396","https://openalex.org/W1987290518","https://openalex.org/W1995227244","https://openalex.org/W2022204379","https://openalex.org/W2041660125","https://openalex.org/W2047839295","https://openalex.org/W2055768543","https://openalex.org/W2061750102","https://openalex.org/W2069356606","https://openalex.org/W2089559088","https://openalex.org/W2100564780","https://openalex.org/W2108305472","https://openalex.org/W2161002641","https://openalex.org/W2751868889","https://openalex.org/W2978329087","https://openalex.org/W4242859866"],"related_works":["https://openalex.org/W1561297735","https://openalex.org/W3080515648","https://openalex.org/W4220840321","https://openalex.org/W4281571022","https://openalex.org/W3197558611","https://openalex.org/W2962964459","https://openalex.org/W2353584259","https://openalex.org/W2886062995","https://openalex.org/W2058164831","https://openalex.org/W2005381510"],"abstract_inverted_index":null,"counts_by_year":[{"year":2026,"cited_by_count":2},{"year":2025,"cited_by_count":14},{"year":2024,"cited_by_count":11},{"year":2023,"cited_by_count":8},{"year":2022,"cited_by_count":10},{"year":2021,"cited_by_count":17},{"year":2020,"cited_by_count":10},{"year":2019,"cited_by_count":6},{"year":2018,"cited_by_count":13},{"year":2017,"cited_by_count":9},{"year":2016,"cited_by_count":3},{"year":2015,"cited_by_count":2},{"year":2014,"cited_by_count":3},{"year":2013,"cited_by_count":3},{"year":2012,"cited_by_count":3}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}