{"id":"https://openalex.org/W2043157166","doi":"https://doi.org/10.1137/130931011","title":"Entropic Approximation for Mathematical Programs with Robust Equilibrium Constraints","display_name":"Entropic Approximation for Mathematical Programs with Robust Equilibrium Constraints","publication_year":2014,"publication_date":"2014-01-01","ids":{"openalex":"https://openalex.org/W2043157166","doi":"https://doi.org/10.1137/130931011","mag":"2043157166"},"language":"en","primary_location":{"id":"doi:10.1137/130931011","is_oa":false,"landing_page_url":"https://doi.org/10.1137/130931011","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":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5100617985","display_name":"Yongchao Liu","orcid":"https://orcid.org/0000-0001-9955-4414"},"institutions":[{"id":"https://openalex.org/I43313876","display_name":"Dalian Maritime University","ror":"https://ror.org/002b7nr53","country_code":"CN","type":"education","lineage":["https://openalex.org/I43313876"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Yongchao Liu","raw_affiliation_strings":["Department of Mathematics, Dalian Maritime University, Dalian 116026, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Dalian Maritime University, Dalian 116026, China","institution_ids":["https://openalex.org/I43313876"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5076597279","display_name":"Huifu Xu","orcid":"https://orcid.org/0000-0001-8307-2920"},"institutions":[{"id":"https://openalex.org/I180825142","display_name":"City, University of London","ror":"https://ror.org/04489at23","country_code":"GB","type":"education","lineage":["https://openalex.org/I124357947","https://openalex.org/I180825142"]}],"countries":["GB"],"is_corresponding":true,"raw_author_name":"Huifu Xu","raw_affiliation_strings":["School of Engineering and Mathematical Sciences, City University London, EC1V 0HB, London, UK"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Engineering and Mathematical Sciences, City University London, EC1V 0HB, London, UK","institution_ids":["https://openalex.org/I180825142"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5076597279"],"corresponding_institution_ids":["https://openalex.org/I180825142"],"apc_list":null,"apc_paid":null,"fwci":2.3485,"has_fulltext":false,"cited_by_count":14,"citation_normalized_percentile":{"value":0.8880991,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":88,"max":98},"biblio":{"volume":"24","issue":"3","first_page":"933","last_page":"958"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11413","display_name":"Risk and Portfolio Optimization","score":0.9991999864578247,"subfield":{"id":"https://openalex.org/subfields/1803","display_name":"Management Science and Operations Research"},"field":{"id":"https://openalex.org/fields/18","display_name":"Decision Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},"topics":[{"id":"https://openalex.org/T11413","display_name":"Risk and Portfolio Optimization","score":0.9991999864578247,"subfield":{"id":"https://openalex.org/subfields/1803","display_name":"Management Science and Operations Research"},"field":{"id":"https://openalex.org/fields/18","display_name":"Decision Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},{"id":"https://openalex.org/T12709","display_name":"Optimization and Mathematical Programming","score":0.9969000220298767,"subfield":{"id":"https://openalex.org/subfields/2207","display_name":"Control and Systems Engineering"},"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/T10969","display_name":"Water resources management and optimization","score":0.9944999814033508,"subfield":{"id":"https://openalex.org/subfields/2212","display_name":"Ocean Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/complementarity","display_name":"Complementarity (molecular biology)","score":0.6608988046646118},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.6099271774291992},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5987389087677002},{"id":"https://openalex.org/keywords/ambiguity","display_name":"Ambiguity","score":0.5593326091766357},{"id":"https://openalex.org/keywords/complementarity-theory","display_name":"Complementarity theory","score":0.5311248302459717},{"id":"https://openalex.org/keywords/probability-measure","display_name":"Probability measure","score":0.506639301776886},{"id":"https://openalex.org/keywords/stochastic-programming","display_name":"Stochastic programming","score":0.4864475429058075},{"id":"https://openalex.org/keywords/expected-value","display_name":"Expected value","score":0.4483986794948578},{"id":"https://openalex.org/keywords/stochastic-approximation","display_name":"Stochastic approximation","score":0.4383275508880615},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.42292261123657227},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.41944754123687744},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.2604410648345947},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.11853864789009094},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.1140279769897461}],"concepts":[{"id":"https://openalex.org/C202269582","wikidata":"https://www.wikidata.org/wiki/Q2644277","display_name":"Complementarity (molecular biology)","level":2,"score":0.6608988046646118},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.6099271774291992},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5987389087677002},{"id":"https://openalex.org/C2780522230","wikidata":"https://www.wikidata.org/wiki/Q1140419","display_name":"Ambiguity","level":2,"score":0.5593326091766357},{"id":"https://openalex.org/C85404239","wikidata":"https://www.wikidata.org/wiki/Q5156417","display_name":"Complementarity theory","level":3,"score":0.5311248302459717},{"id":"https://openalex.org/C21031990","wikidata":"https://www.wikidata.org/wiki/Q355020","display_name":"Probability measure","level":2,"score":0.506639301776886},{"id":"https://openalex.org/C137631369","wikidata":"https://www.wikidata.org/wiki/Q7617831","display_name":"Stochastic programming","level":2,"score":0.4864475429058075},{"id":"https://openalex.org/C141042865","wikidata":"https://www.wikidata.org/wiki/Q200125","display_name":"Expected value","level":2,"score":0.4483986794948578},{"id":"https://openalex.org/C55479107","wikidata":"https://www.wikidata.org/wiki/Q97663916","display_name":"Stochastic approximation","level":3,"score":0.4383275508880615},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.42292261123657227},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.41944754123687744},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.2604410648345947},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.11853864789009094},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.1140279769897461},{"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/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","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/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.0},{"id":"https://openalex.org/C26517878","wikidata":"https://www.wikidata.org/wiki/Q228039","display_name":"Key (lock)","level":2,"score":0.0},{"id":"https://openalex.org/C38652104","wikidata":"https://www.wikidata.org/wiki/Q3510521","display_name":"Computer security","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1137/130931011","is_oa":false,"landing_page_url":"https://doi.org/10.1137/130931011","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:openaccess.city.ac.uk:3267","is_oa":false,"landing_page_url":"https://doi.org/10.1137/130931011>","pdf_url":null,"source":{"id":"https://openalex.org/S4306401940","display_name":"City Research Online (City University London)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I180825142","host_organization_name":"City, University of London","host_organization_lineage":["https://openalex.org/I180825142"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"acceptedVersion","is_accepted":true,"is_published":false,"raw_source_name":null,"raw_type":"Article"},{"id":"pmh:oai:eprints.soton.ac.uk:390732","is_oa":false,"landing_page_url":"https://eprints.soton.ac.uk/390732/","pdf_url":null,"source":{"id":"https://openalex.org/S4306401019","display_name":"ePrints Soton (University of Southampton)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I43439940","host_organization_name":"University of Southampton","host_organization_lineage":["https://openalex.org/I43439940"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Article"}],"best_oa_location":null,"sustainable_development_goals":[{"score":0.47999998927116394,"id":"https://metadata.un.org/sdg/16","display_name":"Peace, Justice and strong institutions"}],"awards":[{"id":"https://openalex.org/G1782754538","display_name":"On a Robust Approach for Stochastic Equilibrium Problems","funder_award_id":"EP/J014427/1","funder_id":"https://openalex.org/F4320334627","funder_display_name":"Engineering and Physical Sciences Research Council"},{"id":"https://openalex.org/G4859279963","display_name":null,"funder_award_id":"EP/J014427/1","funder_id":"https://openalex.org/F4320334627","funder_display_name":"Engineering and Physical Sciences Research Council"}],"funders":[{"id":"https://openalex.org/F4320334627","display_name":"Engineering and Physical Sciences Research Council","ror":"https://ror.org/0439y7842"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":42,"referenced_works":["https://openalex.org/W17229534","https://openalex.org/W217831350","https://openalex.org/W759920449","https://openalex.org/W1484551447","https://openalex.org/W1541527977","https://openalex.org/W1557270667","https://openalex.org/W1601741115","https://openalex.org/W1604627832","https://openalex.org/W1968355947","https://openalex.org/W1969310585","https://openalex.org/W1980247637","https://openalex.org/W1995285162","https://openalex.org/W2014923785","https://openalex.org/W2018851374","https://openalex.org/W2023209926","https://openalex.org/W2024034153","https://openalex.org/W2031279008","https://openalex.org/W2035238076","https://openalex.org/W2041219907","https://openalex.org/W2042016018","https://openalex.org/W2042028954","https://openalex.org/W2042422381","https://openalex.org/W2042838878","https://openalex.org/W2054321814","https://openalex.org/W2061767581","https://openalex.org/W2078135889","https://openalex.org/W2080474145","https://openalex.org/W2082451773","https://openalex.org/W2082477992","https://openalex.org/W2089105401","https://openalex.org/W2100072100","https://openalex.org/W2106254871","https://openalex.org/W2117781171","https://openalex.org/W2119303899","https://openalex.org/W2131295967","https://openalex.org/W2142092335","https://openalex.org/W2167412190","https://openalex.org/W2562747313","https://openalex.org/W3106992008","https://openalex.org/W3124407081","https://openalex.org/W3143187486","https://openalex.org/W4240533965"],"related_works":["https://openalex.org/W2522776430","https://openalex.org/W2551076699","https://openalex.org/W64605562","https://openalex.org/W61315285","https://openalex.org/W2281909915","https://openalex.org/W2950466654","https://openalex.org/W2105819246","https://openalex.org/W2043801859","https://openalex.org/W2096448982","https://openalex.org/W2171244871"],"abstract_inverted_index":{"In":[0],"this":[1],"paper,":[2],"we":[3,45,69,109],"consider":[4],"a":[5,16,25,58,90,114,125],"class":[6,91],"of":[7,18,43,61,75,92],"mathematical":[8,96,115],"programs":[9,97],"with":[10,57,98,103,117,130],"robust":[11,119],"equilibrium":[12,120],"constraints":[13,21,36,56,100,133],"represented":[14],"by":[15,32,135],"system":[17],"semi-infinite":[19,55,131],"complementarity":[20,35,99],"(SICC).":[22],"We":[23],"propose":[24],"numerical":[26],"scheme":[27,85],"for":[28],"tackling":[29],"SICC.":[30],"Specifically,":[31],"relaxing":[33],"the":[34,40,47,54,71,76,80,104,111,145],"and":[37,79],"then":[38,87],"randomizing":[39],"index":[41],"set":[42,78,142],"SICC,":[44],"employ":[46],"well-known":[48],"entropic":[49],"risk":[50],"measure":[51],"to":[52,89,113],"approximate":[53],"finite":[59],"number":[60],"stochastic":[62,95,128,132],"inequality":[63],"constraints.":[64],"Under":[65],"some":[66,136],"moderate":[67],"conditions,":[68],"quantify":[70],"approximation":[72,84],"in":[73,101],"terms":[74],"feasible":[77],"optimal":[81],"value.":[82],"The":[83],"is":[86,123],"applied":[88],"two":[93],"stage":[94,127],"combination":[102],"polynomial":[105],"decision":[106],"rules.":[107],"Finally,":[108],"extend":[110],"discussion":[112],"program":[116,129],"distributionally":[118],"constraints,":[121],"which":[122],"essentially":[124],"one":[126],"indexed":[134],"probability":[137],"measures":[138],"from":[139],"an":[140],"ambiguity":[141],"defined":[143],"through":[144],"KL-divergence.":[146]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":1},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":1},{"year":2018,"cited_by_count":2},{"year":2017,"cited_by_count":4},{"year":2016,"cited_by_count":2}],"updated_date":"2026-08-26T07:47:46.906454","created_date":"2025-10-10T00:00:00"}