{"id":"https://openalex.org/W2098958718","doi":"https://doi.org/10.1137/15m1036543","title":"Exact Algorithms for Linear Matrix Inequalities","display_name":"Exact Algorithms for Linear Matrix Inequalities","publication_year":2016,"publication_date":"2016-01-01","ids":{"openalex":"https://openalex.org/W2098958718","doi":"https://doi.org/10.1137/15m1036543","mag":"2098958718"},"language":"en","primary_location":{"id":"doi:10.1137/15m1036543","is_oa":false,"landing_page_url":"https://doi.org/10.1137/15m1036543","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":"preprint","indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1508.03715","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5074893139","display_name":"Didier Henrion","orcid":"https://orcid.org/0000-0001-6735-7715"},"institutions":[{"id":"https://openalex.org/I44504214","display_name":"Czech Technical University in Prague","ror":"https://ror.org/03kqpb082","country_code":"CZ","type":"education","lineage":["https://openalex.org/I44504214"]}],"countries":["CZ"],"is_corresponding":false,"raw_author_name":"Didier Henrion","raw_affiliation_strings":["Czech Technical University in Prague"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Czech Technical University in Prague","institution_ids":["https://openalex.org/I44504214"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5014288326","display_name":"Simone Naldi","orcid":"https://orcid.org/0000-0002-4556-6935"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Simone Naldi","raw_affiliation_strings":["LAAS-MAC"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"LAAS-MAC","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5048254079","display_name":"Mohab Safey El Din","orcid":"https://orcid.org/0000-0001-9463-1257"},"institutions":[{"id":"https://openalex.org/I4210159731","display_name":"LIP6","ror":"https://ror.org/05krcen59","country_code":"FR","type":"facility","lineage":["https://openalex.org/I1294671590","https://openalex.org/I1294671590","https://openalex.org/I39804081","https://openalex.org/I4210159245","https://openalex.org/I4210159731"]}],"countries":["FR"],"is_corresponding":false,"raw_author_name":"Mohab Safey El Din","raw_affiliation_strings":["LIP6, PolSys"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"LIP6, PolSys","institution_ids":["https://openalex.org/I4210159731"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":1.8013,"has_fulltext":false,"cited_by_count":4,"citation_normalized_percentile":{"value":0.82219472,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":94,"max":96},"biblio":{"volume":"26","issue":"4","first_page":"2512","last_page":"2539"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.9997000098228455,"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":0.9997000098228455,"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/T11435","display_name":"Polynomial and algebraic computation","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/T10792","display_name":"Matrix Theory and Algorithms","score":0.9962999820709229,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7436195611953735},{"id":"https://openalex.org/keywords/interior-point-method","display_name":"Interior point method","score":0.615031361579895},{"id":"https://openalex.org/keywords/pencil","display_name":"Pencil (optics)","score":0.5581272840499878},{"id":"https://openalex.org/keywords/algebraic-number","display_name":"Algebraic number","score":0.5521027445793152},{"id":"https://openalex.org/keywords/rank","display_name":"Rank (graph theory)","score":0.509641170501709},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.49099043011665344},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.49062100052833557},{"id":"https://openalex.org/keywords/matrix-pencil","display_name":"Matrix pencil","score":0.48115456104278564},{"id":"https://openalex.org/keywords/multilinear-map","display_name":"Multilinear map","score":0.48090028762817383},{"id":"https://openalex.org/keywords/degree","display_name":"Degree (music)","score":0.4723939597606659},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4546944200992584},{"id":"https://openalex.org/keywords/quadratic-equation","display_name":"Quadratic equation","score":0.4335235357284546},{"id":"https://openalex.org/keywords/positive-definite-matrix","display_name":"Positive-definite matrix","score":0.42944714426994324},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.3233550488948822},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.19657030701637268},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.08942416310310364}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7436195611953735},{"id":"https://openalex.org/C155253501","wikidata":"https://www.wikidata.org/wiki/Q461992","display_name":"Interior point method","level":2,"score":0.615031361579895},{"id":"https://openalex.org/C134949993","wikidata":"https://www.wikidata.org/wiki/Q2068617","display_name":"Pencil (optics)","level":2,"score":0.5581272840499878},{"id":"https://openalex.org/C9376300","wikidata":"https://www.wikidata.org/wiki/Q168817","display_name":"Algebraic number","level":2,"score":0.5521027445793152},{"id":"https://openalex.org/C164226766","wikidata":"https://www.wikidata.org/wiki/Q7293202","display_name":"Rank (graph theory)","level":2,"score":0.509641170501709},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.49099043011665344},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.49062100052833557},{"id":"https://openalex.org/C1576492","wikidata":"https://www.wikidata.org/wiki/Q3739929","display_name":"Matrix pencil","level":3,"score":0.48115456104278564},{"id":"https://openalex.org/C84392682","wikidata":"https://www.wikidata.org/wiki/Q1952404","display_name":"Multilinear map","level":2,"score":0.48090028762817383},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.4723939597606659},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4546944200992584},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.4335235357284546},{"id":"https://openalex.org/C49712288","wikidata":"https://www.wikidata.org/wiki/Q77601250","display_name":"Positive-definite matrix","level":3,"score":0.42944714426994324},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.3233550488948822},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.19657030701637268},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.08942416310310364},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.0},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.0},{"id":"https://openalex.org/C24890656","wikidata":"https://www.wikidata.org/wiki/Q82811","display_name":"Acoustics","level":1,"score":0.0},{"id":"https://openalex.org/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"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/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0},{"id":"https://openalex.org/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","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/C78519656","wikidata":"https://www.wikidata.org/wiki/Q101333","display_name":"Mechanical engineering","level":1,"score":0.0}],"mesh":[],"locations_count":6,"locations":[{"id":"doi:10.1137/15m1036543","is_oa":false,"landing_page_url":"https://doi.org/10.1137/15m1036543","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:arXiv.org:1508.03715","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1508.03715","pdf_url":"https://arxiv.org/pdf/1508.03715","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":"","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.698.7178","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.698.7178","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://homepages.laas.fr/henrion/Papers/exactlmi.pdf","raw_type":"text"},{"id":"pmh:oai:HAL:hal-01184320v2","is_oa":false,"landing_page_url":"https://hal.science/hal-01184320","pdf_url":null,"source":{"id":"https://openalex.org/S4306402512","display_name":"HAL (Le Centre pour la Communication Scientifique Directe)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1294671590","host_organization_name":"Centre National de la Recherche Scientifique","host_organization_lineage":["https://openalex.org/I1294671590"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"SIAM Journal on Optimization, 2016, 26 (4), pp.2512-2539. &#x27E8;10.1137/15M1036543&#x27E9;","raw_type":"info:eu-repo/semantics/article"},{"id":"doi:10.48550/arxiv.1508.03715","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.1508.03715","pdf_url":null,"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":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"Preprint"},{"id":"mag:2098958718","is_oa":false,"landing_page_url":null,"pdf_url":null,"source":null,"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":null}],"best_oa_location":{"id":"pmh:oai:arXiv.org:1508.03715","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1508.03715","pdf_url":"https://arxiv.org/pdf/1508.03715","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":"","raw_type":"text"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G5356588235","display_name":"GEOmetry and algebra of Linear Matrix Inequalities with control applications","funder_award_id":"ANR-11-BS03-0011","funder_id":"https://openalex.org/F4320320883","funder_display_name":"Agence Nationale de la Recherche"}],"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":88,"referenced_works":["https://openalex.org/W16627279","https://openalex.org/W85128250","https://openalex.org/W99708544","https://openalex.org/W126279018","https://openalex.org/W186221740","https://openalex.org/W618928362","https://openalex.org/W624623585","https://openalex.org/W653291882","https://openalex.org/W1480066506","https://openalex.org/W1495410784","https://openalex.org/W1498202005","https://openalex.org/W1518341038","https://openalex.org/W1559197676","https://openalex.org/W1559760555","https://openalex.org/W1574151794","https://openalex.org/W1577258119","https://openalex.org/W1590002045","https://openalex.org/W1632601927","https://openalex.org/W1668282207","https://openalex.org/W1828839365","https://openalex.org/W1877476807","https://openalex.org/W1964100506","https://openalex.org/W1964603869","https://openalex.org/W1967184028","https://openalex.org/W1967344706","https://openalex.org/W1973993014","https://openalex.org/W1979387186","https://openalex.org/W1985924702","https://openalex.org/W1986124084","https://openalex.org/W1988025893","https://openalex.org/W1989923040","https://openalex.org/W1991813962","https://openalex.org/W2000714860","https://openalex.org/W2001255903","https://openalex.org/W2003232014","https://openalex.org/W2011693299","https://openalex.org/W2011936157","https://openalex.org/W2016403269","https://openalex.org/W2018701055","https://openalex.org/W2024340911","https://openalex.org/W2033016032","https://openalex.org/W2038345254","https://openalex.org/W2039539620","https://openalex.org/W2059522106","https://openalex.org/W2060414131","https://openalex.org/W2062162367","https://openalex.org/W2063194632","https://openalex.org/W2064131066","https://openalex.org/W2065053468","https://openalex.org/W2065235897","https://openalex.org/W2065706751","https://openalex.org/W2078365883","https://openalex.org/W2082387287","https://openalex.org/W2083449425","https://openalex.org/W2100440346","https://openalex.org/W2104223485","https://openalex.org/W2112086696","https://openalex.org/W2117819179","https://openalex.org/W2124706092","https://openalex.org/W2128687423","https://openalex.org/W2130918732","https://openalex.org/W2137508197","https://openalex.org/W2151549243","https://openalex.org/W2164838307","https://openalex.org/W2165281529","https://openalex.org/W2166998199","https://openalex.org/W2170042437","https://openalex.org/W2212277022","https://openalex.org/W2238531909","https://openalex.org/W2268784238","https://openalex.org/W2279788372","https://openalex.org/W2288091890","https://openalex.org/W2912522929","https://openalex.org/W2912833207","https://openalex.org/W2949730746","https://openalex.org/W2962712276","https://openalex.org/W2962767810","https://openalex.org/W2962819657","https://openalex.org/W2963967210","https://openalex.org/W2964297942","https://openalex.org/W3098762489","https://openalex.org/W3102900599","https://openalex.org/W3104810083","https://openalex.org/W3210839039","https://openalex.org/W3217399827","https://openalex.org/W4233252806","https://openalex.org/W4254742099","https://openalex.org/W4255442211"],"related_works":["https://openalex.org/W2962893120","https://openalex.org/W2785875198","https://openalex.org/W2011936157","https://openalex.org/W2170042437","https://openalex.org/W2212277022","https://openalex.org/W2964003194","https://openalex.org/W3023736450","https://openalex.org/W1491431298","https://openalex.org/W1964181492","https://openalex.org/W1852930718","https://openalex.org/W2789189302","https://openalex.org/W2900621127","https://openalex.org/W2065235897","https://openalex.org/W3012581685","https://openalex.org/W2273701522","https://openalex.org/W2963548828","https://openalex.org/W3198950050","https://openalex.org/W2962920905","https://openalex.org/W2963547774","https://openalex.org/W3157546919"],"abstract_inverted_index":{"Let":[0],"$A(x)":[1],"=":[2],"A_0+x_1A_1+\\cdots+x_nA_n$":[3],"be":[4],"a":[5,36,44,122,155,168],"linear":[6,45],"matrix,":[7],"or":[8,75],"pencil,":[9],"generated":[10],"by":[11,43,182],"given":[12],"symmetric":[13],"matrices":[14],"$A_0,A_1,\\ldots,A_n$":[15],"of":[16,24,67,88,99,108,112,121,137,145,159,185],"size":[17],"$m$":[18,162],"with":[19,117,190],"rational":[20],"entries.":[21],"The":[22,81,105],"set":[23,39],"real":[25],"vectors":[26],"$x$":[27],"such":[28,167],"that":[29,77,141,148],"the":[30,59,73,86,93,97,100,103,109,113,118,128,135,142,172],"pencil":[31,101],"is":[32,35,79,151,165,174],"positive":[33],"semidefinite":[34,124],"convex":[37],"semialgebraic":[38],"called":[40],"spectrahedron,":[41,74],"described":[42],"matrix":[46],"inequality.":[47],"We":[48,130,180],"design":[49],"an":[50,63,89],"exact":[51,64],"algorithm":[52,82],"that,":[53],"up":[54],"to":[55,127,192],"genericity":[56],"assumptions":[57],"on":[58,102],"input":[60],"matrices,":[61],"computes":[62],"algebraic":[65,110,119],"representation":[66,111],"at":[68],"least":[69],"one":[70],"point":[71,95,114],"in":[72,154,176],"decides":[76],"it":[78],"empty.":[80],"does":[83],"not":[84],"assume":[85],"existence":[87],"interior":[90],"point,":[91],"and":[92,170],"computed":[94],"minimizes":[96],"rank":[98],"spectrahedron.":[104],"degree":[106,120],"$d$":[107],"coincides":[115],"experimentally":[116],"generic":[123],"program":[125],"associated":[126],"pencil.":[129],"provide":[131],"explicit":[132],"bounds":[133],"for":[134],"complexity":[136],"our":[138],"algorithm,":[139],"proving":[140],"maximum":[143],"number":[144],"arithmetic":[146],"operations":[147],"are":[149],"performed":[150],"essentially":[152],"quadratic":[153],"multilinear":[156],"B\u00e9zout":[157],"bound":[158],"$d$.":[160],"When":[161],"(resp.,":[163,178],"$n$)":[164],"fixed,":[166],"bound,":[169],"hence":[171],"complexity,":[173],"polynomial":[175],"$n$":[177],"$m$).":[179],"conclude":[181],"providing":[183],"results":[184],"experiments":[186],"showing":[187],"practical":[188],"improvements":[189],"respect":[191],"state-of-the-art":[193],"computer":[194],"algebra":[195],"algorithms.":[196]},"counts_by_year":[{"year":2017,"cited_by_count":2},{"year":2016,"cited_by_count":2}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}
