{"id":"https://openalex.org/W1966150378","doi":"https://doi.org/10.1137/s1052623493252985","title":"Analysis of a Symmetric Rank-One Trust Region Method","display_name":"Analysis of a Symmetric Rank-One Trust Region Method","publication_year":1996,"publication_date":"1996-11-01","ids":{"openalex":"https://openalex.org/W1966150378","doi":"https://doi.org/10.1137/s1052623493252985","mag":"1966150378"},"language":"en","primary_location":{"id":"doi:10.1137/s1052623493252985","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s1052623493252985","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/A5110661221","display_name":"Richard H. Byrd","orcid":null},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Richard H. Byrd","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5050018842","display_name":"Humaid Khalfan","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Humaid Fayez Khalfan","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5002119487","display_name":"Robert B. Schnabel","orcid":"https://orcid.org/0000-0003-1707-0294"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Robert B. Schnabel","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5110661221"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.9637,"has_fulltext":false,"cited_by_count":64,"citation_normalized_percentile":{"value":0.71650663,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":99},"biblio":{"volume":"6","issue":"4","first_page":"1025","last_page":"1039"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.9998999834060669,"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":0.9998999834060669,"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/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.9973999857902527,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"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/T10792","display_name":"Matrix Theory and Algorithms","score":0.9937999844551086,"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/trust-region","display_name":"Trust region","score":0.9091702103614807},{"id":"https://openalex.org/keywords/hessian-matrix","display_name":"Hessian matrix","score":0.8206133842468262},{"id":"https://openalex.org/keywords/iterated-function","display_name":"Iterated function","score":0.7470492124557495},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6945414543151855},{"id":"https://openalex.org/keywords/rank","display_name":"Rank (graph theory)","score":0.6573865413665771},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.6066113114356995},{"id":"https://openalex.org/keywords/independence","display_name":"Independence (probability theory)","score":0.5860509276390076},{"id":"https://openalex.org/keywords/definiteness","display_name":"Definiteness","score":0.5020875930786133},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.48762527108192444},{"id":"https://openalex.org/keywords/quasi-newton-method","display_name":"Quasi-Newton method","score":0.4833959639072418},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.4748133420944214},{"id":"https://openalex.org/keywords/feature","display_name":"Feature (linguistics)","score":0.46227604150772095},{"id":"https://openalex.org/keywords/rate-of-convergence","display_name":"Rate of convergence","score":0.4507560133934021},{"id":"https://openalex.org/keywords/positive-definiteness","display_name":"Positive definiteness","score":0.44918474555015564},{"id":"https://openalex.org/keywords/newtons-method","display_name":"Newton's method","score":0.34186333417892456},{"id":"https://openalex.org/keywords/positive-definite-matrix","display_name":"Positive-definite matrix","score":0.30717718601226807},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.24578037858009338},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.2168956696987152},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.12459906935691833},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.11394533514976501},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.111583411693573},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.06213352084159851}],"concepts":[{"id":"https://openalex.org/C89109886","wikidata":"https://www.wikidata.org/wiki/Q1535924","display_name":"Trust region","level":3,"score":0.9091702103614807},{"id":"https://openalex.org/C203616005","wikidata":"https://www.wikidata.org/wiki/Q620495","display_name":"Hessian matrix","level":2,"score":0.8206133842468262},{"id":"https://openalex.org/C140479938","wikidata":"https://www.wikidata.org/wiki/Q5254619","display_name":"Iterated function","level":2,"score":0.7470492124557495},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6945414543151855},{"id":"https://openalex.org/C164226766","wikidata":"https://www.wikidata.org/wiki/Q7293202","display_name":"Rank (graph theory)","level":2,"score":0.6573865413665771},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.6066113114356995},{"id":"https://openalex.org/C35651441","wikidata":"https://www.wikidata.org/wiki/Q625303","display_name":"Independence (probability theory)","level":2,"score":0.5860509276390076},{"id":"https://openalex.org/C2779056149","wikidata":"https://www.wikidata.org/wiki/Q1182686","display_name":"Definiteness","level":2,"score":0.5020875930786133},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.48762527108192444},{"id":"https://openalex.org/C114954040","wikidata":"https://www.wikidata.org/wiki/Q1476018","display_name":"Quasi-Newton method","level":4,"score":0.4833959639072418},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.4748133420944214},{"id":"https://openalex.org/C2776401178","wikidata":"https://www.wikidata.org/wiki/Q12050496","display_name":"Feature (linguistics)","level":2,"score":0.46227604150772095},{"id":"https://openalex.org/C57869625","wikidata":"https://www.wikidata.org/wiki/Q1783502","display_name":"Rate of convergence","level":3,"score":0.4507560133934021},{"id":"https://openalex.org/C2778265155","wikidata":"https://www.wikidata.org/wiki/Q7233276","display_name":"Positive definiteness","level":4,"score":0.44918474555015564},{"id":"https://openalex.org/C85189116","wikidata":"https://www.wikidata.org/wiki/Q374195","display_name":"Newton's method","level":3,"score":0.34186333417892456},{"id":"https://openalex.org/C49712288","wikidata":"https://www.wikidata.org/wiki/Q77601250","display_name":"Positive-definite matrix","level":3,"score":0.30717718601226807},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.24578037858009338},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.2168956696987152},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.12459906935691833},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.11394533514976501},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.111583411693573},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.06213352084159851},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C38652104","wikidata":"https://www.wikidata.org/wiki/Q3510521","display_name":"Computer security","level":1,"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/C127162648","wikidata":"https://www.wikidata.org/wiki/Q16858953","display_name":"Channel (broadcasting)","level":2,"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/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","level":1,"score":0.0},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"id":"https://openalex.org/C178635117","wikidata":"https://www.wikidata.org/wiki/Q747499","display_name":"RADIUS","level":2,"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/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/s1052623493252985","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s1052623493252985","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"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":12,"referenced_works":["https://openalex.org/W1485207068","https://openalex.org/W1579664581","https://openalex.org/W1992976319","https://openalex.org/W1996723024","https://openalex.org/W2048221927","https://openalex.org/W2053964895","https://openalex.org/W2056603712","https://openalex.org/W2060217404","https://openalex.org/W2075184363","https://openalex.org/W2084236346","https://openalex.org/W2095211439","https://openalex.org/W4299512035"],"related_works":["https://openalex.org/W2774735586","https://openalex.org/W2383147277","https://openalex.org/W2369793817","https://openalex.org/W2380066421","https://openalex.org/W2364489971","https://openalex.org/W2361979571","https://openalex.org/W4205226845","https://openalex.org/W3181246138","https://openalex.org/W4210436583","https://openalex.org/W2087978208"],"abstract_inverted_index":{"Several":[0],"recent":[1,78],"computational":[2,114],"studies":[3],"have":[4,73],"shown":[5],"that":[6,38,72,87,94,99,116,118,125],"the":[7,39,56,63,69,97,100,129,135,138],"symmetric":[8],"rank-one":[9],"(SR1)":[10],"update":[11,17],"is":[12,88,90,103,126],"a":[13,24,28,123],"very":[14],"competitive":[15],"quasi-Newton":[16],"in":[18,76,122],"optimization":[19,35],"algorithms.":[20],"This":[21],"paper":[22],"gives":[23],"new":[25],"analysis":[26,52],"of":[27,49,55,58,62,68,80,137],"trust":[29,84,140],"region":[30,85,141],"SR1":[31,81,139],"method":[32,40,86],"for":[33],"unconstrained":[34],"and":[36],"shows":[37],"has":[41],"an":[42],"$n":[43],"+":[44],"1$":[45],"step":[46],"q-superlinear":[47],"rate":[48],"convergence.":[50],"The":[51,83],"makes":[53],"neither":[54],"assumptions":[57],"uniform":[59],"linear":[60],"independence":[61],"iterates":[64],"nor":[65],"positive":[66],"definiteness":[67],"Hessian":[70,101],"approximations":[71],"been":[74],"made":[75],"other":[77],"analyses":[79],"methods.":[82],"analyzed":[89],"fairly":[91],"standard,":[92],"except":[93],"it":[95],"includes":[96],"feature":[98],"approximation":[102],"updated":[104],"after":[105],"all":[106],"steps,":[107],"including":[108],"rejected":[109],"steps.":[110],"We":[111],"also":[112],"present":[113],"results":[115],"show":[117],"this":[119],"feature,":[120],"safeguarded":[121],"way":[124],"consistent":[127],"with":[128],"convergence":[130],"analysis,":[131],"does":[132],"not":[133],"harm":[134],"efficiency":[136],"method.":[142]},"counts_by_year":[{"year":2026,"cited_by_count":2},{"year":2025,"cited_by_count":4},{"year":2024,"cited_by_count":4},{"year":2023,"cited_by_count":6},{"year":2022,"cited_by_count":2},{"year":2021,"cited_by_count":8},{"year":2020,"cited_by_count":3},{"year":2019,"cited_by_count":2},{"year":2018,"cited_by_count":5},{"year":2017,"cited_by_count":1},{"year":2016,"cited_by_count":2},{"year":2015,"cited_by_count":2},{"year":2014,"cited_by_count":4}],"updated_date":"2026-04-13T07:58:08.660418","created_date":"2025-10-10T00:00:00"}
