{"id":"https://openalex.org/W2759377581","doi":"https://doi.org/10.1137/16m1095056","title":"On the Generalized Lanczos Trust-Region Method","display_name":"On the Generalized Lanczos Trust-Region Method","publication_year":2017,"publication_date":"2017-01-01","ids":{"openalex":"https://openalex.org/W2759377581","doi":"https://doi.org/10.1137/16m1095056","mag":"2759377581"},"language":"en","primary_location":{"id":"doi:10.1137/16m1095056","is_oa":false,"landing_page_url":"https://doi.org/10.1137/16m1095056","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/A5102019567","display_name":"Lei\u2010Hong Zhang","orcid":"https://orcid.org/0000-0001-5349-8621"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Lei-Hong Zhang","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5060270577","display_name":"Chungen Shen","orcid":"https://orcid.org/0000-0002-0391-5033"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Chungen Shen","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5011252539","display_name":"Ren\u2010Cang Li","orcid":"https://orcid.org/0000-0002-4388-3398"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Ren-Cang Li","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":3,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":3.1104,"has_fulltext":false,"cited_by_count":31,"citation_normalized_percentile":{"value":0.92751358,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":95,"max":99},"biblio":{"volume":"27","issue":"3","first_page":"2110","last_page":"2142"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9997000098228455,"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"}},"topics":[{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9997000098228455,"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/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/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.9984999895095825,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/trust-region","display_name":"Trust region","score":0.8452029228210449},{"id":"https://openalex.org/keywords/solver","display_name":"Solver","score":0.7918762564659119},{"id":"https://openalex.org/keywords/lanczos-resampling","display_name":"Lanczos resampling","score":0.7002982497215271},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.6335632801055908},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6153846979141235},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.5424476861953735},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.5358316898345947},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4535147547721863},{"id":"https://openalex.org/keywords/a-priori-and-a-posteriori","display_name":"A priori and a posteriori","score":0.4527280330657959},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.443718820810318},{"id":"https://openalex.org/keywords/lanczos-algorithm","display_name":"Lanczos algorithm","score":0.4215920567512512},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.3873435854911804},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.3707587718963623},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.08437660336494446},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.066403329372406}],"concepts":[{"id":"https://openalex.org/C89109886","wikidata":"https://www.wikidata.org/wiki/Q1535924","display_name":"Trust region","level":3,"score":0.8452029228210449},{"id":"https://openalex.org/C2778770139","wikidata":"https://www.wikidata.org/wiki/Q1966904","display_name":"Solver","level":2,"score":0.7918762564659119},{"id":"https://openalex.org/C119256216","wikidata":"https://www.wikidata.org/wiki/Q913012","display_name":"Lanczos resampling","level":3,"score":0.7002982497215271},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.6335632801055908},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6153846979141235},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.5424476861953735},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.5358316898345947},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4535147547721863},{"id":"https://openalex.org/C75553542","wikidata":"https://www.wikidata.org/wiki/Q178161","display_name":"A priori and a posteriori","level":2,"score":0.4527280330657959},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.443718820810318},{"id":"https://openalex.org/C20501136","wikidata":"https://www.wikidata.org/wiki/Q366640","display_name":"Lanczos algorithm","level":4,"score":0.4215920567512512},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.3873435854911804},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3707587718963623},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.08437660336494446},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.066403329372406},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","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/C111472728","wikidata":"https://www.wikidata.org/wiki/Q9471","display_name":"Epistemology","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/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","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/C50522688","wikidata":"https://www.wikidata.org/wiki/Q189833","display_name":"Economic growth","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/C38652104","wikidata":"https://www.wikidata.org/wiki/Q3510521","display_name":"Computer security","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/16m1095056","is_oa":false,"landing_page_url":"https://doi.org/10.1137/16m1095056","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":[{"id":"https://openalex.org/G2283596320","display_name":null,"funder_award_id":"11428104","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G4073502623","display_name":null,"funder_award_id":"NSFC-11271259","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G5288410646","display_name":null,"funder_award_id":"NSFC-11371102","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G5907499065","display_name":null,"funder_award_id":"CCF-1527104","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G5951886075","display_name":null,"funder_award_id":"NSFC-11101281","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G6066137772","display_name":null,"funder_award_id":"NSFC-11671246","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G762094221","display_name":null,"funder_award_id":"DMS-1317330","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320324781","display_name":"Shanghai University of Finance and Economics","ror":"https://ror.org/00wtvfq62"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":29,"referenced_works":["https://openalex.org/W1563853427","https://openalex.org/W1760551737","https://openalex.org/W1964831500","https://openalex.org/W1968988636","https://openalex.org/W1974888421","https://openalex.org/W1976798627","https://openalex.org/W1979603710","https://openalex.org/W1981674498","https://openalex.org/W1994666506","https://openalex.org/W2000389939","https://openalex.org/W2017279613","https://openalex.org/W2026079992","https://openalex.org/W2029277933","https://openalex.org/W2058885431","https://openalex.org/W2059164920","https://openalex.org/W2063690908","https://openalex.org/W2075184363","https://openalex.org/W2080641403","https://openalex.org/W2080851285","https://openalex.org/W2086079401","https://openalex.org/W2101734906","https://openalex.org/W2143866207","https://openalex.org/W2146295302","https://openalex.org/W2149454052","https://openalex.org/W2157725068","https://openalex.org/W2160064536","https://openalex.org/W3003775594","https://openalex.org/W3135067840","https://openalex.org/W4285719527"],"related_works":["https://openalex.org/W2156009096","https://openalex.org/W3176451595","https://openalex.org/W2027838537","https://openalex.org/W1970812947","https://openalex.org/W2058241770","https://openalex.org/W1980579360","https://openalex.org/W4248073715","https://openalex.org/W4243541541","https://openalex.org/W2095282848","https://openalex.org/W2550228252"],"abstract_inverted_index":{"The":[0,45,219],"so-called":[1],"trust-region":[2,9,48,87,211],"subproblem":[3],"gets":[4],"its":[5,101],"name":[6],"in":[7,11,19,29,104,200,229],"the":[8,30,77,82,86,95,130,134,141,173,178,181,186,191,196,201,210,215,225,231],"method":[10,50,80],"optimization":[12],"and":[13,61,106,144,154,184,208],"also":[14],"plays":[15],"a":[16,73,125],"vital":[17],"role":[18],"various":[20],"other":[21],"applications.":[22],"Several":[23],"numerical":[24,120,168,220],"algorithms":[25],"have":[26],"been":[27],"proposed":[28,51],"literature":[31],"for":[32,41,81,118,129,159,185],"solving":[33],"small-to-medium":[34],"size":[35],"dense":[36],"problems":[37],"as":[38,40,138,140,156,163,204],"well":[39,139],"large-scale":[42],"sparse":[43],"problems.":[44],"generalized":[46],"Lanczos":[47,79],"(GLTR)":[49],"by":[52],"[N.":[53],"I.":[54],"M.":[55,59],"Gould,":[56],"S.":[57],"Lucidi,":[58],"Roma":[60],"P.":[62],"L.":[63],"Toint,":[64],"SIAM":[65],"J.":[66],"Optim.,":[67],"9":[68],"(1999),":[69],"pp.":[70],"561--580]":[71],"is":[72],"natural":[74],"extension":[75],"of":[76,97,167,180,234],"classical":[78],"linear":[83],"system":[84],"to":[85,99,132],"subproblem.":[88],"In":[89,172],"this":[90],"paper,":[91],"we":[92,123,176,189],"first":[93,174],"analyze":[94],"convergence":[96,102,131],"GLTR":[98,117,199],"reveal":[100],"behavior":[103],"theory":[105],"then":[107],"propose":[108],"new":[109,205,226],"stopping":[110,157,206,227],"criteria":[111,158,207,228],"that":[112,146],"can":[113,149,236],"be":[114,150,237],"integrated":[115],"into":[116,195],"better":[119],"performance.":[121],"Specifically,":[122],"develop":[124],"priori":[126],"upper":[127,182,192],"bounds":[128,148],"both":[133],"optimal":[135,142],"objective":[136],"value":[137],"solution":[143],"argue":[145],"these":[147],"efficiently":[151],"estimated":[152],"numerically":[153],"serve":[155],"iterative":[160],"methods":[161],"such":[162],"GLTR.":[164],"Two":[165],"sets":[166],"tests":[169],"are":[170],"presented.":[171],"set,":[175,188],"demonstrate":[177],"sharpness":[179],"bounds,":[183],"second":[187],"integrate":[190],"bound":[193],"estimate":[194],"Fortran":[197],"routine":[198],"library":[202],"GALAHAD":[203],"test":[209],"solver":[212],"TRU":[213,235],"on":[214],"problem":[216],"collection":[217],"CUTEr.":[218],"results":[221],"show":[222],"that,":[223],"with":[224],"\\textttGLTR,":[230],"overall":[232],"performance":[233],"improved":[238],"considerably.":[239]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":5},{"year":2023,"cited_by_count":3},{"year":2022,"cited_by_count":4},{"year":2021,"cited_by_count":4},{"year":2020,"cited_by_count":3},{"year":2019,"cited_by_count":7},{"year":2018,"cited_by_count":3}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}