{"id":"https://openalex.org/W2028602808","doi":"https://doi.org/10.1007/s10898-009-9484-1","title":"On convex relaxations of quadrilinear terms","display_name":"On convex relaxations of quadrilinear terms","publication_year":2009,"publication_date":"2009-11-09","ids":{"openalex":"https://openalex.org/W2028602808","doi":"https://doi.org/10.1007/s10898-009-9484-1","mag":"2028602808"},"language":"en","primary_location":{"id":"doi:10.1007/s10898-009-9484-1","is_oa":false,"landing_page_url":"https://doi.org/10.1007/s10898-009-9484-1","pdf_url":null,"source":{"id":"https://openalex.org/S115925768","display_name":"Journal of Global Optimization","issn_l":"0925-5001","issn":["0925-5001","1573-2916"],"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":"Journal of Global 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/A5025519046","display_name":"Sonia Cafieri","orcid":"https://orcid.org/0000-0002-4578-8394"},"institutions":[{"id":"https://openalex.org/I142476485","display_name":"\u00c9cole Polytechnique","ror":"https://ror.org/05hy3tk52","country_code":"FR","type":"education","lineage":["https://openalex.org/I142476485","https://openalex.org/I4210145102"]},{"id":"https://openalex.org/I4210139461","display_name":"Laboratoire d'Informatique de l'\u00c9cole Polytechnique","ror":"https://ror.org/04afed728","country_code":"FR","type":"facility","lineage":["https://openalex.org/I1294671590","https://openalex.org/I1294671590","https://openalex.org/I1326498283","https://openalex.org/I142476485","https://openalex.org/I4210139461","https://openalex.org/I4210145102","https://openalex.org/I4210159245"]}],"countries":["FR"],"is_corresponding":true,"raw_author_name":"Sonia Cafieri","raw_affiliation_strings":["LIX, \u00c9cole Polytechnique, Palaiseau, France","LIX, \u00c9cole Polytechnique, Palaiseau, France 91128#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"LIX, \u00c9cole Polytechnique, Palaiseau, France","institution_ids":["https://openalex.org/I4210139461"]},{"raw_affiliation_string":"LIX, \u00c9cole Polytechnique, Palaiseau, France 91128#TAB#","institution_ids":["https://openalex.org/I142476485"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5031373908","display_name":"Jon Lee","orcid":"https://orcid.org/0000-0002-8190-1091"},"institutions":[{"id":"https://openalex.org/I1341412227","display_name":"IBM (United States)","ror":"https://ror.org/05hh8d621","country_code":"US","type":"company","lineage":["https://openalex.org/I1341412227"]},{"id":"https://openalex.org/I4210114115","display_name":"IBM Research - Thomas J. Watson Research Center","ror":"https://ror.org/0265w5591","country_code":"US","type":"facility","lineage":["https://openalex.org/I1341412227","https://openalex.org/I4210114115"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Jon Lee","raw_affiliation_strings":["Department of Mathematical Sciences, IBM T.J. Watson Research Center, Yorktown Heights, USA","Department of Mathematical Sciences, IBM T.J. Watson Research Center, Yorktown Heights, USA 10598"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematical Sciences, IBM T.J. Watson Research Center, Yorktown Heights, USA","institution_ids":["https://openalex.org/I4210114115"]},{"raw_affiliation_string":"Department of Mathematical Sciences, IBM T.J. Watson Research Center, Yorktown Heights, USA 10598","institution_ids":["https://openalex.org/I1341412227"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5046138085","display_name":"Leo Liberti","orcid":"https://orcid.org/0000-0003-3139-6821"},"institutions":[{"id":"https://openalex.org/I142476485","display_name":"\u00c9cole Polytechnique","ror":"https://ror.org/05hy3tk52","country_code":"FR","type":"education","lineage":["https://openalex.org/I142476485","https://openalex.org/I4210145102"]},{"id":"https://openalex.org/I4210139461","display_name":"Laboratoire d'Informatique de l'\u00c9cole Polytechnique","ror":"https://ror.org/04afed728","country_code":"FR","type":"facility","lineage":["https://openalex.org/I1294671590","https://openalex.org/I1294671590","https://openalex.org/I1326498283","https://openalex.org/I142476485","https://openalex.org/I4210139461","https://openalex.org/I4210145102","https://openalex.org/I4210159245"]}],"countries":["FR"],"is_corresponding":true,"raw_author_name":"Leo Liberti","raw_affiliation_strings":["LIX, \u00c9cole Polytechnique, Palaiseau, France","LIX, \u00c9cole Polytechnique, Palaiseau, France 91128#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"LIX, \u00c9cole Polytechnique, Palaiseau, France","institution_ids":["https://openalex.org/I4210139461"]},{"raw_affiliation_string":"LIX, \u00c9cole Polytechnique, Palaiseau, France 91128#TAB#","institution_ids":["https://openalex.org/I142476485"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5025519046","https://openalex.org/A5031373908","https://openalex.org/A5046138085"],"corresponding_institution_ids":["https://openalex.org/I1341412227","https://openalex.org/I142476485","https://openalex.org/I4210114115","https://openalex.org/I4210139461"],"apc_list":{"value":2780,"currency":"USD","value_usd":2780},"apc_paid":null,"fwci":6.956,"has_fulltext":false,"cited_by_count":54,"citation_normalized_percentile":{"value":0.97699021,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"47","issue":"4","first_page":"661","last_page":"685"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.9983000159263611,"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"}},"topics":[{"id":"https://openalex.org/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.9983000159263611,"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/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.9907000064849854,"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/T12288","display_name":"Optimization and Search Problems","score":0.9728999733924866,"subfield":{"id":"https://openalex.org/subfields/1705","display_name":"Computer Networks and Communications"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8328086733818054},{"id":"https://openalex.org/keywords/bilinear-interpolation","display_name":"Bilinear interpolation","score":0.7759765386581421},{"id":"https://openalex.org/keywords/relaxation","display_name":"Relaxation (psychology)","score":0.559110701084137},{"id":"https://openalex.org/keywords/regular-polygon","display_name":"Regular polygon","score":0.49299928545951843},{"id":"https://openalex.org/keywords/convex-optimization","display_name":"Convex optimization","score":0.4859675168991089},{"id":"https://openalex.org/keywords/multilinear-map","display_name":"Multilinear map","score":0.46447718143463135},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.4635699391365051},{"id":"https://openalex.org/keywords/convex-analysis","display_name":"Convex analysis","score":0.43654486536979675},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.4157165586948395},{"id":"https://openalex.org/keywords/envelope","display_name":"Envelope (radar)","score":0.4143926203250885},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.397690087556839},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.33678463101387024},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.3241513967514038},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.23082426190376282},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.1307390034198761},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.08331531286239624}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8328086733818054},{"id":"https://openalex.org/C205203396","wikidata":"https://www.wikidata.org/wiki/Q612143","display_name":"Bilinear interpolation","level":2,"score":0.7759765386581421},{"id":"https://openalex.org/C2776029896","wikidata":"https://www.wikidata.org/wiki/Q3935810","display_name":"Relaxation (psychology)","level":2,"score":0.559110701084137},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.49299928545951843},{"id":"https://openalex.org/C157972887","wikidata":"https://www.wikidata.org/wiki/Q463359","display_name":"Convex optimization","level":3,"score":0.4859675168991089},{"id":"https://openalex.org/C84392682","wikidata":"https://www.wikidata.org/wiki/Q1952404","display_name":"Multilinear map","level":2,"score":0.46447718143463135},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.4635699391365051},{"id":"https://openalex.org/C12108790","wikidata":"https://www.wikidata.org/wiki/Q2234833","display_name":"Convex analysis","level":4,"score":0.43654486536979675},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.4157165586948395},{"id":"https://openalex.org/C65155139","wikidata":"https://www.wikidata.org/wiki/Q5380912","display_name":"Envelope (radar)","level":3,"score":0.4143926203250885},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.397690087556839},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.33678463101387024},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3241513967514038},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.23082426190376282},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.1307390034198761},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.08331531286239624},{"id":"https://openalex.org/C77805123","wikidata":"https://www.wikidata.org/wiki/Q161272","display_name":"Social psychology","level":1,"score":0.0},{"id":"https://openalex.org/C15744967","wikidata":"https://www.wikidata.org/wiki/Q9418","display_name":"Psychology","level":0,"score":0.0},{"id":"https://openalex.org/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","level":1,"score":0.0},{"id":"https://openalex.org/C554190296","wikidata":"https://www.wikidata.org/wiki/Q47528","display_name":"Radar","level":2,"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/C76155785","wikidata":"https://www.wikidata.org/wiki/Q418","display_name":"Telecommunications","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/s10898-009-9484-1","is_oa":false,"landing_page_url":"https://doi.org/10.1007/s10898-009-9484-1","pdf_url":null,"source":{"id":"https://openalex.org/S115925768","display_name":"Journal of Global Optimization","issn_l":"0925-5001","issn":["0925-5001","1573-2916"],"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":"Journal of Global Optimization","raw_type":"journal-article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.157.9599","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.157.9599","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.lix.polytechnique.fr/~liberti/quartic.pdf","raw_type":"text"},{"id":"pmh:oai:RePEc:spr:jglopt:v:47:y:2010:i:4:p:661-685","is_oa":false,"landing_page_url":"http://hdl.handle.net/10.1007/s10898-009-9484-1","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":[{"score":0.4399999976158142,"display_name":"Reduced inequalities","id":"https://metadata.un.org/sdg/10"}],"awards":[],"funders":[{"id":"https://openalex.org/F4320320883","display_name":"Agence Nationale de la Recherche","ror":"https://ror.org/00rbzpz17"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":45,"referenced_works":["https://openalex.org/W40231621","https://openalex.org/W40319279","https://openalex.org/W44187364","https://openalex.org/W51788365","https://openalex.org/W52628724","https://openalex.org/W54695997","https://openalex.org/W178571860","https://openalex.org/W600585552","https://openalex.org/W639273726","https://openalex.org/W1482203794","https://openalex.org/W1482298744","https://openalex.org/W1497465005","https://openalex.org/W1540276704","https://openalex.org/W1564380039","https://openalex.org/W1582520000","https://openalex.org/W1591699350","https://openalex.org/W1596630133","https://openalex.org/W1825288209","https://openalex.org/W1984425167","https://openalex.org/W1988622494","https://openalex.org/W1998221953","https://openalex.org/W2001741445","https://openalex.org/W2003862318","https://openalex.org/W2022484305","https://openalex.org/W2023374550","https://openalex.org/W2023672624","https://openalex.org/W2026063499","https://openalex.org/W2032234281","https://openalex.org/W2034307786","https://openalex.org/W2039992864","https://openalex.org/W2048881327","https://openalex.org/W2058851899","https://openalex.org/W2066580436","https://openalex.org/W2068054962","https://openalex.org/W2073605101","https://openalex.org/W2084006722","https://openalex.org/W2130183935","https://openalex.org/W2132750992","https://openalex.org/W2293000863","https://openalex.org/W2526264695","https://openalex.org/W2607230115","https://openalex.org/W2770560911","https://openalex.org/W2914693551","https://openalex.org/W4235025507","https://openalex.org/W6635258235"],"related_works":["https://openalex.org/W2056337104","https://openalex.org/W2187499728","https://openalex.org/W4377007384","https://openalex.org/W4250865487","https://openalex.org/W2079893325","https://openalex.org/W4287773735","https://openalex.org/W3187930658","https://openalex.org/W3160071949","https://openalex.org/W3032713827","https://openalex.org/W4287184329"],"abstract_inverted_index":null,"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":1},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":2},{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":4},{"year":2017,"cited_by_count":4},{"year":2016,"cited_by_count":3},{"year":2015,"cited_by_count":7},{"year":2014,"cited_by_count":6},{"year":2013,"cited_by_count":4},{"year":2012,"cited_by_count":12}],"updated_date":"2026-08-08T01:25:22.217667","created_date":"2025-10-10T00:00:00"}
