{"id":"https://openalex.org/W4294957954","doi":"https://doi.org/10.3390/sym14091882","title":"A Two-Step Iteration Method for Vertical Linear Complementarity Problems","display_name":"A Two-Step Iteration Method for Vertical Linear Complementarity Problems","publication_year":2022,"publication_date":"2022-09-08","ids":{"openalex":"https://openalex.org/W4294957954","doi":"https://doi.org/10.3390/sym14091882"},"language":"en","primary_location":{"id":"doi:10.3390/sym14091882","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14091882","pdf_url":"https://www.mdpi.com/2073-8994/14/9/1882/pdf?version=1662635886","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2073-8994/14/9/1882/pdf?version=1662635886","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5015270146","display_name":"Yunlin Song","orcid":null},"institutions":[{"id":"https://openalex.org/I111950717","display_name":"Macau University of Science and Technology","ror":"https://ror.org/03jqs2n27","country_code":"MO","type":"education","lineage":["https://openalex.org/I111950717","https://openalex.org/I4391767947"]}],"countries":["MO"],"is_corresponding":false,"raw_author_name":"Yunlin Song","raw_affiliation_strings":["Department of Engineering Science, Macau University of Sciences and Technology, Macao 999078, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Engineering Science, Macau University of Sciences and Technology, Macao 999078, China","institution_ids":["https://openalex.org/I111950717"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5031618633","display_name":"Hua Zheng","orcid":"https://orcid.org/0000-0002-2822-0078"},"institutions":[{"id":"https://openalex.org/I3130751423","display_name":"Shaoguan University","ror":"https://ror.org/0286g6711","country_code":"CN","type":"education","lineage":["https://openalex.org/I3130751423"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Hua Zheng","raw_affiliation_strings":["School of Mathematics and Statistics, Shaoguan University, Shaoguan 512005, China"],"raw_orcid":"https://orcid.org/0000-0002-2822-0078","affiliations":[{"raw_affiliation_string":"School of Mathematics and Statistics, Shaoguan University, Shaoguan 512005, China","institution_ids":["https://openalex.org/I3130751423"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5090615044","display_name":"Xiaoping Lu","orcid":"https://orcid.org/0000-0002-2363-4175"},"institutions":[{"id":"https://openalex.org/I111950717","display_name":"Macau University of Science and Technology","ror":"https://ror.org/03jqs2n27","country_code":"MO","type":"education","lineage":["https://openalex.org/I111950717","https://openalex.org/I4391767947"]}],"countries":["MO"],"is_corresponding":false,"raw_author_name":"Xiaoping Lu","raw_affiliation_strings":["School of Computer Science and Engineering, Macau University of Sciences and Technology, Macao 999078, China"],"raw_orcid":"https://orcid.org/0000-0002-2363-4175","affiliations":[{"raw_affiliation_string":"School of Computer Science and Engineering, Macau University of Sciences and Technology, Macao 999078, China","institution_ids":["https://openalex.org/I111950717"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5020932623","display_name":"Seakweng Vong","orcid":"https://orcid.org/0000-0002-3017-3346"},"institutions":[{"id":"https://openalex.org/I204512498","display_name":"University of Macau","ror":"https://ror.org/01r4q9n85","country_code":"MO","type":"education","lineage":["https://openalex.org/I204512498"]}],"countries":["MO"],"is_corresponding":false,"raw_author_name":"Seak-Weng Vong","raw_affiliation_strings":["Department of Mathematics, University of Macau, Macao 519000, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Macau, Macao 519000, China","institution_ids":["https://openalex.org/I204512498"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5031618633"],"corresponding_institution_ids":["https://openalex.org/I3130751423"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":0.7618,"has_fulltext":true,"cited_by_count":5,"citation_normalized_percentile":{"value":0.74719778,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":94,"max":97},"biblio":{"volume":"14","issue":"9","first_page":"1882","last_page":"1882"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9994999766349792,"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/T10792","display_name":"Matrix Theory and Algorithms","score":0.9994999766349792,"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":0.9983999729156494,"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/T12661","display_name":"Iterative Methods for Nonlinear Equations","score":0.9930999875068665,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/complementarity","display_name":"Complementarity (molecular biology)","score":0.748559832572937},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.6077706813812256},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5955780148506165},{"id":"https://openalex.org/keywords/linear-complementarity-problem","display_name":"Linear complementarity problem","score":0.5724048614501953},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.5637879371643066},{"id":"https://openalex.org/keywords/mixed-complementarity-problem","display_name":"Mixed complementarity problem","score":0.5465835928916931},{"id":"https://openalex.org/keywords/complementarity-theory","display_name":"Complementarity theory","score":0.5303295850753784},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.48084935545921326},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.4596627950668335},{"id":"https://openalex.org/keywords/numerical-analysis","display_name":"Numerical analysis","score":0.4300185739994049},{"id":"https://openalex.org/keywords/iterative-method","display_name":"Iterative method","score":0.41978245973587036},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.34781613945961},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.18471163511276245},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.11504185199737549}],"concepts":[{"id":"https://openalex.org/C202269582","wikidata":"https://www.wikidata.org/wiki/Q2644277","display_name":"Complementarity (molecular biology)","level":2,"score":0.748559832572937},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.6077706813812256},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5955780148506165},{"id":"https://openalex.org/C34320192","wikidata":"https://www.wikidata.org/wiki/Q783105","display_name":"Linear complementarity problem","level":3,"score":0.5724048614501953},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.5637879371643066},{"id":"https://openalex.org/C77196050","wikidata":"https://www.wikidata.org/wiki/Q6883962","display_name":"Mixed complementarity problem","level":3,"score":0.5465835928916931},{"id":"https://openalex.org/C85404239","wikidata":"https://www.wikidata.org/wiki/Q5156417","display_name":"Complementarity theory","level":3,"score":0.5303295850753784},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.48084935545921326},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.4596627950668335},{"id":"https://openalex.org/C48753275","wikidata":"https://www.wikidata.org/wiki/Q11216","display_name":"Numerical analysis","level":2,"score":0.4300185739994049},{"id":"https://openalex.org/C159694833","wikidata":"https://www.wikidata.org/wiki/Q2321565","display_name":"Iterative method","level":2,"score":0.41978245973587036},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.34781613945961},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.18471163511276245},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.11504185199737549},{"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},{"id":"https://openalex.org/C54355233","wikidata":"https://www.wikidata.org/wiki/Q7162","display_name":"Genetics","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/C50522688","wikidata":"https://www.wikidata.org/wiki/Q189833","display_name":"Economic growth","level":1,"score":0.0},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/sym14091882","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14091882","pdf_url":"https://www.mdpi.com/2073-8994/14/9/1882/pdf?version=1662635886","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:c06a48a42e4c466cb228eb9d35abb055","is_oa":true,"landing_page_url":"https://doaj.org/article/c06a48a42e4c466cb228eb9d35abb055","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry, Vol 14, Iss 9, p 1882 (2022)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2073-8994/14/9/1882/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/sym14091882","pdf_url":null,"source":{"id":"https://openalex.org/S4306400947","display_name":"MDPI (MDPI AG)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210097602","host_organization_name":"Multidisciplinary Digital Publishing Institute (Switzerland)","host_organization_lineage":["https://openalex.org/I4210097602"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/sym14091882","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14091882","pdf_url":"https://www.mdpi.com/2073-8994/14/9/1882/pdf?version=1662635886","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G1460853621","display_name":null,"funder_award_id":"2021KCXTD052","funder_id":"https://openalex.org/F4320335919","funder_display_name":"Major Projects of Guangdong Education Department for Foundation Research and Applied Research"},{"id":"https://openalex.org/G270709787","display_name":null,"funder_award_id":"MYRG2020-00035-FST","funder_id":"https://openalex.org/F4320322841","funder_display_name":"Universidade de Macau"},{"id":"https://openalex.org/G2791469631","display_name":null,"funder_award_id":"SY2021KJ09","funder_id":"https://openalex.org/F4320335919","funder_display_name":"Major Projects of Guangdong Education Department for Foundation Research and Applied Research"},{"id":"https://openalex.org/G3316443298","display_name":null,"funder_award_id":"MYRG2018-00047-FST","funder_id":"https://openalex.org/F4320322841","funder_display_name":"Universidade de Macau"},{"id":"https://openalex.org/G4449064821","display_name":null,"funder_award_id":"MYRG2018-00047-FST","funder_id":"https://openalex.org/F4320335919","funder_display_name":"Major Projects of Guangdong Education Department for Foundation Research and Applied Research"},{"id":"https://openalex.org/G4746046253","display_name":null,"funder_award_id":"2018KZDXM065","funder_id":"https://openalex.org/F4320335919","funder_display_name":"Major Projects of Guangdong Education Department for Foundation Research and Applied Research"},{"id":"https://openalex.org/G5932056203","display_name":null,"funder_award_id":"SZ2020KJ01","funder_id":"https://openalex.org/F4320335919","funder_display_name":"Major Projects of Guangdong Education Department for Foundation Research and Applied Research"},{"id":"https://openalex.org/G804569234","display_name":null,"funder_award_id":"0005/2019/A","funder_id":"https://openalex.org/F4320335919","funder_display_name":"Major Projects of Guangdong Education Department for Foundation Research and Applied Research"},{"id":"https://openalex.org/G8387899994","display_name":null,"funder_award_id":"210716094530390","funder_id":"https://openalex.org/F4320335919","funder_display_name":"Major Projects of Guangdong Education Department for Foundation Research and Applied Research"},{"id":"https://openalex.org/G8854968487","display_name":null,"funder_award_id":"0073/2019/A2","funder_id":"https://openalex.org/F4320335919","funder_display_name":"Major Projects of Guangdong Education Department for Foundation Research and Applied Research"},{"id":"https://openalex.org/G8886074789","display_name":null,"funder_award_id":"MYRG2020-00035-FST","funder_id":"https://openalex.org/F4320335919","funder_display_name":"Major Projects of Guangdong Education Department for Foundation Research and Applied Research"}],"funders":[{"id":"https://openalex.org/F4320322841","display_name":"Universidade de Macau","ror":"https://ror.org/01r4q9n85"},{"id":"https://openalex.org/F4320335919","display_name":"Major Projects of Guangdong Education Department for Foundation Research and Applied Research","ror":null}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4294957954.pdf","grobid_xml":"https://content.openalex.org/works/W4294957954.grobid-xml"},"referenced_works_count":23,"referenced_works":["https://openalex.org/W1963924275","https://openalex.org/W1977117914","https://openalex.org/W1979337419","https://openalex.org/W1982337446","https://openalex.org/W2013321893","https://openalex.org/W2030459733","https://openalex.org/W2063686441","https://openalex.org/W2066977799","https://openalex.org/W2087353196","https://openalex.org/W2093993556","https://openalex.org/W2107517414","https://openalex.org/W2160211628","https://openalex.org/W2165124442","https://openalex.org/W2750706968","https://openalex.org/W2803130912","https://openalex.org/W2890434649","https://openalex.org/W2954877738","https://openalex.org/W3030990663","https://openalex.org/W3082642720","https://openalex.org/W3195563679","https://openalex.org/W4205909128","https://openalex.org/W4286689066","https://openalex.org/W4295016751"],"related_works":["https://openalex.org/W2276648117","https://openalex.org/W2063320018","https://openalex.org/W2042813889","https://openalex.org/W2358901572","https://openalex.org/W2028943836","https://openalex.org/W4220847489","https://openalex.org/W2091810828","https://openalex.org/W2765841216","https://openalex.org/W2599483915","https://openalex.org/W2607337407"],"abstract_inverted_index":{"In":[0],"this":[1],"paper,":[2],"a":[3,14],"two-step":[4,61],"iteration":[5],"method":[6,32,56],"was":[7],"established":[8],"which":[9,35],"can":[10,36],"be":[11],"viewed":[12],"as":[13],"generalisation":[15],"of":[16,29,41,67],"the":[17,30,38,42,47,54,60,65,68],"existing":[18],"modulus-based":[19],"methods":[20],"for":[21],"vertical":[22],"linear":[23],"complementarity":[24],"problems.":[25],"The":[26],"convergence":[27,39],"analysis":[28],"proposed":[31,55],"is":[33,57],"presented,":[34],"enlarge":[37],"domain":[40],"parameter":[43],"matrix":[44],"compared":[45],"to":[46],"recent":[48],"results.":[49,70],"Numerical":[50],"examples":[51],"show":[52],"that":[53],"efficient":[58],"with":[59],"technique":[62],"and":[63],"confirm":[64],"improvement":[66],"theoretical":[69]},"counts_by_year":[{"year":2024,"cited_by_count":3},{"year":2023,"cited_by_count":2}],"updated_date":"2026-05-06T08:25:59.206177","created_date":"2025-10-10T00:00:00"}