{"id":"https://openalex.org/W2007976113","doi":"https://doi.org/10.1137/s1052623493246288","title":"Convergence of Newton's Method for Singular Smooth and Nonsmooth Equations Using Adaptive Outer Inverses","display_name":"Convergence of Newton's Method for Singular Smooth and Nonsmooth Equations Using Adaptive Outer Inverses","publication_year":1997,"publication_date":"1997-05-01","ids":{"openalex":"https://openalex.org/W2007976113","doi":"https://doi.org/10.1137/s1052623493246288","mag":"2007976113"},"language":"en","primary_location":{"id":"doi:10.1137/s1052623493246288","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s1052623493246288","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/A5100369445","display_name":"Xiaojun Chen","orcid":"https://orcid.org/0000-0002-6627-2637"},"institutions":[{"id":"https://openalex.org/I31746571","display_name":"UNSW Sydney","ror":"https://ror.org/03r8z3t63","country_code":"AU","type":"education","lineage":["https://openalex.org/I31746571"]}],"countries":["AU"],"is_corresponding":true,"raw_author_name":"Xiaojun Chen","raw_affiliation_strings":["University Of New south Wales"],"affiliations":[{"raw_affiliation_string":"University Of New south Wales","institution_ids":["https://openalex.org/I31746571"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5078141425","display_name":"Zuhair Nashed","orcid":null},"institutions":[{"id":"https://openalex.org/I86501945","display_name":"University of Delaware","ror":"https://ror.org/01sbq1a82","country_code":"US","type":"education","lineage":["https://openalex.org/I86501945"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Zuhair Nashed","raw_affiliation_strings":["University of Delaware"],"affiliations":[{"raw_affiliation_string":"University of Delaware","institution_ids":["https://openalex.org/I86501945"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5011886900","display_name":"Liqun Qi","orcid":"https://orcid.org/0000-0002-1112-5250"},"institutions":[{"id":"https://openalex.org/I31746571","display_name":"UNSW Sydney","ror":"https://ror.org/03r8z3t63","country_code":"AU","type":"education","lineage":["https://openalex.org/I31746571"]}],"countries":["AU"],"is_corresponding":false,"raw_author_name":"Liqun Qi","raw_affiliation_strings":["University Of New south Wales"],"affiliations":[{"raw_affiliation_string":"University Of New south Wales","institution_ids":["https://openalex.org/I31746571"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5100369445"],"corresponding_institution_ids":["https://openalex.org/I31746571"],"apc_list":null,"apc_paid":null,"fwci":4.0906,"has_fulltext":false,"cited_by_count":63,"citation_normalized_percentile":{"value":0.94674556,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"7","issue":"2","first_page":"445","last_page":"462"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12661","display_name":"Iterative Methods for Nonlinear Equations","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/T12661","display_name":"Iterative Methods for Nonlinear Equations","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/T10792","display_name":"Matrix Theory and Algorithms","score":0.9987000226974487,"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":0.9980000257492065,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8992956280708313},{"id":"https://openalex.org/keywords/lipschitz-continuity","display_name":"Lipschitz continuity","score":0.6826293468475342},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.5339673161506653},{"id":"https://openalex.org/keywords/invertible-matrix","display_name":"Invertible matrix","score":0.5149263143539429},{"id":"https://openalex.org/keywords/bounded-function","display_name":"Bounded function","score":0.5076686143875122},{"id":"https://openalex.org/keywords/jacobian-matrix-and-determinant","display_name":"Jacobian matrix and determinant","score":0.48409998416900635},{"id":"https://openalex.org/keywords/local-convergence","display_name":"Local convergence","score":0.46273669600486755},{"id":"https://openalex.org/keywords/operator","display_name":"Operator (biology)","score":0.4413682520389557},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.4325202703475952},{"id":"https://openalex.org/keywords/newtons-method","display_name":"Newton's method","score":0.4274984896183014},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.3445175290107727},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.343679815530777},{"id":"https://openalex.org/keywords/iterative-method","display_name":"Iterative method","score":0.21920821070671082},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.08954614400863647}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8992956280708313},{"id":"https://openalex.org/C22324862","wikidata":"https://www.wikidata.org/wiki/Q652707","display_name":"Lipschitz continuity","level":2,"score":0.6826293468475342},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.5339673161506653},{"id":"https://openalex.org/C96442724","wikidata":"https://www.wikidata.org/wiki/Q242188","display_name":"Invertible matrix","level":2,"score":0.5149263143539429},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.5076686143875122},{"id":"https://openalex.org/C200331156","wikidata":"https://www.wikidata.org/wiki/Q506041","display_name":"Jacobian matrix and determinant","level":2,"score":0.48409998416900635},{"id":"https://openalex.org/C117898588","wikidata":"https://www.wikidata.org/wiki/Q6664310","display_name":"Local convergence","level":3,"score":0.46273669600486755},{"id":"https://openalex.org/C17020691","wikidata":"https://www.wikidata.org/wiki/Q139677","display_name":"Operator (biology)","level":5,"score":0.4413682520389557},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.4325202703475952},{"id":"https://openalex.org/C85189116","wikidata":"https://www.wikidata.org/wiki/Q374195","display_name":"Newton's method","level":3,"score":0.4274984896183014},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3445175290107727},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.343679815530777},{"id":"https://openalex.org/C159694833","wikidata":"https://www.wikidata.org/wiki/Q2321565","display_name":"Iterative method","level":2,"score":0.21920821070671082},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.08954614400863647},{"id":"https://openalex.org/C55493867","wikidata":"https://www.wikidata.org/wiki/Q7094","display_name":"Biochemistry","level":1,"score":0.0},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","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/C86339819","wikidata":"https://www.wikidata.org/wiki/Q407384","display_name":"Transcription factor","level":3,"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/C104317684","wikidata":"https://www.wikidata.org/wiki/Q7187","display_name":"Gene","level":2,"score":0.0},{"id":"https://openalex.org/C158448853","wikidata":"https://www.wikidata.org/wiki/Q425218","display_name":"Repressor","level":4,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/s1052623493246288","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s1052623493246288","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":32,"referenced_works":["https://openalex.org/W237657096","https://openalex.org/W1968850685","https://openalex.org/W1970551061","https://openalex.org/W1972867892","https://openalex.org/W1976890616","https://openalex.org/W1982027431","https://openalex.org/W1992688650","https://openalex.org/W1996143958","https://openalex.org/W2018169702","https://openalex.org/W2020432186","https://openalex.org/W2036312021","https://openalex.org/W2038178878","https://openalex.org/W2038843658","https://openalex.org/W2047417907","https://openalex.org/W2050864977","https://openalex.org/W2052885339","https://openalex.org/W2053643774","https://openalex.org/W2054204520","https://openalex.org/W2055576996","https://openalex.org/W2056254398","https://openalex.org/W2065691568","https://openalex.org/W2067547116","https://openalex.org/W2068484625","https://openalex.org/W2070890393","https://openalex.org/W2090963365","https://openalex.org/W2092855431","https://openalex.org/W2097720652","https://openalex.org/W2107710567","https://openalex.org/W2113734366","https://openalex.org/W2122434654","https://openalex.org/W2166161964","https://openalex.org/W2343404102"],"related_works":["https://openalex.org/W162626055","https://openalex.org/W3040723915","https://openalex.org/W1901133132","https://openalex.org/W1896162644","https://openalex.org/W2127159799","https://openalex.org/W4287724094","https://openalex.org/W2068966571","https://openalex.org/W2741623878","https://openalex.org/W2083618095","https://openalex.org/W3197573213"],"abstract_inverted_index":{"We":[0,23],"present":[1],"a":[2,27,34,50,60,76,150,156,167],"local":[3],"convergence":[4,102],"analysis":[5],"of":[6,20,30,54,70,79,131,152,158,162],"generalized":[7,77,136],"Newton":[8],"methods":[9],"for":[10,26,40],"singular":[11],"smooth":[12,109,153],"and":[13,110,155],"nonsmooth":[14,159],"operator":[15,63,107,114],"equations":[16,154],"using":[17],"adaptive":[18],"constructs":[19],"outer":[21],"inverses.":[22],"prove":[24,99],"that":[25,39,64,100,125],"solution":[28,51],"$x^*$":[29,89,91],"$F(x)=0$,":[31],"there":[32],"exists":[33],"ball":[35],"$S=S(x^*,r)$,":[36],"$r>0$":[37],"such":[38],"any":[41],"starting":[42],"point":[43],"$x_0\\in":[44],"S$":[45,53],"the":[46,67,101,106,113,123,129],"method":[47],"converges":[48],"to":[49,138,166],"$\\bar{x}^*\\in":[52],"$\\Gamma":[55],"F(x)=0$,":[56],"where":[57],"$\\Gamma$":[58],"is":[59,92,103,108,115,164],"bounded":[61],"linear":[62],"depends":[65],"on":[66,75],"Fr\u00e9chet":[68],"derivative":[69,134,137],"F":[71,80],"at":[72,81],"$x_0$":[73],"or":[74,135],"Jacobian":[78],"$x_0$.":[82],"Point":[83],"$\\bar{x}^*$":[84],"may":[85],"be":[86],"different":[87],"from":[88],"when":[90],"not":[93],"an":[94,132],"isolated":[95],"solution.":[96],"Moreover,":[97],"we":[98],"quadratic":[104],"if":[105,112],"superlinear":[111],"locally":[116],"Lipschitz.":[117],"These":[118],"results":[119,146],"are":[120,147],"sharp":[121],"in":[122,128],"sense":[124],"they":[126],"reduce":[127],"case":[130],"invertible":[133],"earlier":[139],"theorems":[140],"with":[141],"no":[142],"additional":[143],"assumptions.":[144],"The":[145],"illustrated":[148],"by":[149],"system":[151,157],"equations,":[160],"each":[161],"which":[163],"equivalent":[165],"nonlinear":[168],"complementarity":[169],"problem.":[170]},"counts_by_year":[{"year":2022,"cited_by_count":2},{"year":2021,"cited_by_count":2},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":3},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":2},{"year":2016,"cited_by_count":6},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":3},{"year":2013,"cited_by_count":5},{"year":2012,"cited_by_count":5}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}