{"id":"https://openalex.org/W2020357931","doi":"https://doi.org/10.1155/2014/324181","title":"Optimal Algorithms and the BFGS Updating Techniques for Solving Unconstrained Nonlinear Minimization Problems","display_name":"Optimal Algorithms and the BFGS Updating Techniques for Solving Unconstrained Nonlinear Minimization Problems","publication_year":2014,"publication_date":"2014-01-01","ids":{"openalex":"https://openalex.org/W2020357931","doi":"https://doi.org/10.1155/2014/324181","mag":"2020357931"},"language":"en","primary_location":{"id":"doi:10.1155/2014/324181","is_oa":true,"landing_page_url":"https://doi.org/10.1155/2014/324181","pdf_url":"https://downloads.hindawi.com/journals/jam/2014/324181.pdf","source":{"id":"https://openalex.org/S190082376","display_name":"Journal of Applied Mathematics","issn_l":"1110-757X","issn":["1110-757X","1687-0042"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310319869","host_organization_name":"Hindawi Publishing Corporation","host_organization_lineage":["https://openalex.org/P4310319869","https://openalex.org/P4310320595"],"host_organization_lineage_names":["Hindawi Publishing Corporation","Wiley"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Applied Mathematics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://downloads.hindawi.com/journals/jam/2014/324181.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5090276712","display_name":"Chein\u2010Shan Liu","orcid":"https://orcid.org/0000-0001-6366-3539"},"institutions":[{"id":"https://openalex.org/I16733864","display_name":"National Taiwan University","ror":"https://ror.org/05bqach95","country_code":"TW","type":"education","lineage":["https://openalex.org/I16733864"]}],"countries":["TW"],"is_corresponding":true,"raw_author_name":"Chein-Shan Liu","raw_affiliation_strings":["Department of Civil Engineering, National Taiwan University, Taipei 106-17, Taiwan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Civil Engineering, National Taiwan University, Taipei 106-17, Taiwan","institution_ids":["https://openalex.org/I16733864"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5090276712"],"corresponding_institution_ids":["https://openalex.org/I16733864"],"apc_list":{"value":1130,"currency":"USD","value_usd":1130},"apc_paid":{"value":1130,"currency":"USD","value_usd":1130},"fwci":1.4122,"has_fulltext":true,"cited_by_count":6,"citation_normalized_percentile":{"value":0.79142116,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":97},"biblio":{"volume":"2014","issue":null,"first_page":"1","last_page":"14"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.9998000264167786,"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":0.9998000264167786,"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":0.9962999820709229,"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/T10100","display_name":"Metaheuristic Optimization Algorithms Research","score":0.9617999792098999,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"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/broyden\u2013fletcher\u2013goldfarb\u2013shanno-algorithm","display_name":"Broyden\u2013Fletcher\u2013Goldfarb\u2013Shanno algorithm","score":0.9130114316940308},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.6542603373527527},{"id":"https://openalex.org/keywords/gradient-descent","display_name":"Gradient descent","score":0.6146115660667419},{"id":"https://openalex.org/keywords/descent-direction","display_name":"Descent direction","score":0.5380956530570984},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.5234367847442627},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5194756388664246},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.5129446387290955},{"id":"https://openalex.org/keywords/manifold","display_name":"Manifold (fluid mechanics)","score":0.5125305652618408},{"id":"https://openalex.org/keywords/minification","display_name":"Minification","score":0.5121414661407471},{"id":"https://openalex.org/keywords/metric","display_name":"Metric (unit)","score":0.5012979507446289},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.44887685775756836},{"id":"https://openalex.org/keywords/quadratic-equation","display_name":"Quadratic equation","score":0.44195476174354553},{"id":"https://openalex.org/keywords/rank","display_name":"Rank (graph theory)","score":0.4248325526714325},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.3732587695121765},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.09888747334480286},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.08126333355903625}],"concepts":[{"id":"https://openalex.org/C132721684","wikidata":"https://www.wikidata.org/wiki/Q2877013","display_name":"Broyden\u2013Fletcher\u2013Goldfarb\u2013Shanno algorithm","level":3,"score":0.9130114316940308},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.6542603373527527},{"id":"https://openalex.org/C153258448","wikidata":"https://www.wikidata.org/wiki/Q1199743","display_name":"Gradient descent","level":3,"score":0.6146115660667419},{"id":"https://openalex.org/C116149140","wikidata":"https://www.wikidata.org/wiki/Q2070951","display_name":"Descent direction","level":4,"score":0.5380956530570984},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.5234367847442627},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5194756388664246},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.5129446387290955},{"id":"https://openalex.org/C529865628","wikidata":"https://www.wikidata.org/wiki/Q1790740","display_name":"Manifold (fluid mechanics)","level":2,"score":0.5125305652618408},{"id":"https://openalex.org/C147764199","wikidata":"https://www.wikidata.org/wiki/Q6865248","display_name":"Minification","level":2,"score":0.5121414661407471},{"id":"https://openalex.org/C176217482","wikidata":"https://www.wikidata.org/wiki/Q860554","display_name":"Metric (unit)","level":2,"score":0.5012979507446289},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.44887685775756836},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.44195476174354553},{"id":"https://openalex.org/C164226766","wikidata":"https://www.wikidata.org/wiki/Q7293202","display_name":"Rank (graph theory)","level":2,"score":0.4248325526714325},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3732587695121765},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.09888747334480286},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.08126333355903625},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"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/C78519656","wikidata":"https://www.wikidata.org/wiki/Q101333","display_name":"Mechanical engineering","level":1,"score":0.0},{"id":"https://openalex.org/C31258907","wikidata":"https://www.wikidata.org/wiki/Q1301371","display_name":"Computer network","level":1,"score":0.0},{"id":"https://openalex.org/C21547014","wikidata":"https://www.wikidata.org/wiki/Q1423657","display_name":"Operations management","level":1,"score":0.0},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.0},{"id":"https://openalex.org/C151319957","wikidata":"https://www.wikidata.org/wiki/Q752739","display_name":"Asynchronous communication","level":2,"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/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"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/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1155/2014/324181","is_oa":true,"landing_page_url":"https://doi.org/10.1155/2014/324181","pdf_url":"https://downloads.hindawi.com/journals/jam/2014/324181.pdf","source":{"id":"https://openalex.org/S190082376","display_name":"Journal of Applied Mathematics","issn_l":"1110-757X","issn":["1110-757X","1687-0042"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310319869","host_organization_name":"Hindawi Publishing Corporation","host_organization_lineage":["https://openalex.org/P4310319869","https://openalex.org/P4310320595"],"host_organization_lineage_names":["Hindawi Publishing Corporation","Wiley"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Applied Mathematics","raw_type":"journal-article"},{"id":"pmh:oai:CULeuclid:euclid.jam/1425305598","is_oa":false,"landing_page_url":"https://projecteuclid.org/euclid.jam/1425305598","pdf_url":null,"source":{"id":"https://openalex.org/S4306400787","display_name":"Project Euclid (Cornell University)","issn_l":null,"issn":null,"is_oa":false,"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:RePEc:hin:jnljam:324181","is_oa":false,"landing_page_url":"http://downloads.hindawi.com/journals/JAM/2014/324181.xml","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"},{"id":"pmh:oai:doaj.org/article:3d6fbcd61699436ebc11d237380b6826","is_oa":true,"landing_page_url":"https://doaj.org/article/3d6fbcd61699436ebc11d237380b6826","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":"Journal of Applied Mathematics, Vol 2014 (2014)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.1155/2014/324181","is_oa":true,"landing_page_url":"https://doi.org/10.1155/2014/324181","pdf_url":"https://downloads.hindawi.com/journals/jam/2014/324181.pdf","source":{"id":"https://openalex.org/S190082376","display_name":"Journal of Applied Mathematics","issn_l":"1110-757X","issn":["1110-757X","1687-0042"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310319869","host_organization_name":"Hindawi Publishing Corporation","host_organization_lineage":["https://openalex.org/P4310319869","https://openalex.org/P4310320595"],"host_organization_lineage_names":["Hindawi Publishing Corporation","Wiley"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Applied Mathematics","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G3147666796","display_name":null,"funder_award_id":"102-2221-E-002-125-MY3","funder_id":"https://openalex.org/F4320321040","funder_display_name":"National Science Council"},{"id":"https://openalex.org/G54589887","display_name":null,"funder_award_id":"NSC-102-2221-E-002-125-MY3","funder_id":"https://openalex.org/F4320321040","funder_display_name":"National Science Council"}],"funders":[{"id":"https://openalex.org/F4320321040","display_name":"National Science Council","ror":"https://ror.org/02kv4zf79"},{"id":"https://openalex.org/F4320323900","display_name":"National Taiwan University","ror":"https://ror.org/05bqach95"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2020357931.pdf","grobid_xml":"https://content.openalex.org/works/W2020357931.grobid-xml"},"referenced_works_count":34,"referenced_works":["https://openalex.org/W61118079","https://openalex.org/W173320072","https://openalex.org/W1497361805","https://openalex.org/W1497566274","https://openalex.org/W1530695493","https://openalex.org/W1981556389","https://openalex.org/W1982067912","https://openalex.org/W1991750265","https://openalex.org/W1996669831","https://openalex.org/W1999446604","https://openalex.org/W2005136695","https://openalex.org/W2012341322","https://openalex.org/W2022772618","https://openalex.org/W2026258334","https://openalex.org/W2038210983","https://openalex.org/W2039771911","https://openalex.org/W2042803051","https://openalex.org/W2045010238","https://openalex.org/W2050096287","https://openalex.org/W2056641273","https://openalex.org/W2070473094","https://openalex.org/W2076605490","https://openalex.org/W2078409719","https://openalex.org/W2080894690","https://openalex.org/W2085694061","https://openalex.org/W2090192300","https://openalex.org/W2092087339","https://openalex.org/W2116538711","https://openalex.org/W2141102526","https://openalex.org/W2152074179","https://openalex.org/W2156273800","https://openalex.org/W2339458331","https://openalex.org/W2989128237","https://openalex.org/W4285719527"],"related_works":["https://openalex.org/W2077886840","https://openalex.org/W3199158697","https://openalex.org/W1983625179","https://openalex.org/W2136780394","https://openalex.org/W3101185163","https://openalex.org/W2002303885","https://openalex.org/W4297883503","https://openalex.org/W2615612939","https://openalex.org/W1993453088","https://openalex.org/W1169901681"],"abstract_inverted_index":{"To":[0],"solve":[1],"an":[2,9,37],"unconstrained":[3],"nonlinear":[4],"minimization":[5],"problem,":[6],"we":[7],"propose":[8],"optimal":[10,18,38],"algorithm":[11,19,65],"(OA)":[12],"as":[13,15,86],"well":[14],"a":[16,54,62],"globally":[17],"(GOA),":[20],"by":[21],"deflecting":[22],"the":[23,27,48,67,72,90,96,113,120],"gradient":[24],"direction":[25,30],"to":[26,60],"best":[28],"descent":[29,92],"at":[31],"each":[32],"iteration":[33],"step,":[34],"and":[35,66,95,106,119,127],"with":[36,108],"parameter":[39],"being":[40,87],"derived":[41],"explicitly.":[42],"An":[43],"invariant":[44],"manifold":[45],"defined":[46],"for":[47],"model":[49],"problem":[50],"in":[51,82],"terms":[52],"of":[53,76,116],"locally":[55],"quadratic":[56],"function":[57],"is":[58,69],"used":[59],"derive":[61],"purely":[63],"iterative":[64],"convergence":[68],"proven.":[70],"Then,":[71],"rank-two":[73],"updating":[74],"techniques":[75],"BFGS":[77],"are":[78,104],"employed,":[79],"which":[80],"result":[81],"several":[83],"novel":[84],"algorithms":[85,115],"faster":[88],"than":[89],"steepest":[91],"method":[93,99],"(SDM)":[94],"variable":[97],"metric":[98],"(DFP).":[100],"Six":[101],"numerical":[102],"examples":[103],"examined":[105],"compared":[107],"exact":[109],"solutions,":[110],"revealing":[111],"that":[112],"new":[114],"OA,":[117],"GOA,":[118],"updated":[121],"ones":[122],"have":[123],"superior":[124],"computational":[125],"efficiency":[126],"accuracy.":[128]},"counts_by_year":[{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":1},{"year":2018,"cited_by_count":1},{"year":2016,"cited_by_count":3}],"updated_date":"2026-08-21T09:56:20.448147","created_date":"2025-10-10T00:00:00"}
