{"id":"https://openalex.org/W1597581062","doi":"https://doi.org/10.1137/110830150","title":"Convergence of the Restricted Nelder--Mead Algorithm in Two Dimensions","display_name":"Convergence of the Restricted Nelder--Mead Algorithm in Two Dimensions","publication_year":2012,"publication_date":"2012-01-01","ids":{"openalex":"https://openalex.org/W1597581062","doi":"https://doi.org/10.1137/110830150","mag":"1597581062"},"language":"en","primary_location":{"id":"doi:10.1137/110830150","is_oa":false,"landing_page_url":"https://doi.org/10.1137/110830150","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":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1104.0350","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5005612207","display_name":"Jeffrey C. Lagarias","orcid":"https://orcid.org/0000-0002-4157-5527"},"institutions":[{"id":"https://openalex.org/I27837315","display_name":"University of Michigan","ror":"https://ror.org/00jmfr291","country_code":"US","type":"education","lineage":["https://openalex.org/I27837315"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Jeffrey C. Lagarias","raw_affiliation_strings":["University of Michigan , Ann Arbor"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Michigan , Ann Arbor","institution_ids":["https://openalex.org/I27837315"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5034540754","display_name":"Bjorn Poonen","orcid":"https://orcid.org/0000-0002-8593-2792"},"institutions":[{"id":"https://openalex.org/I63966007","display_name":"Massachusetts Institute of Technology","ror":"https://ror.org/042nb2s44","country_code":"US","type":"education","lineage":["https://openalex.org/I63966007"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Bjorn Poonen","raw_affiliation_strings":["Massachusetts Institute of Technology"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Massachusetts Institute of Technology","institution_ids":["https://openalex.org/I63966007"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5103458692","display_name":"Margaret H. Wright","orcid":null},"institutions":[{"id":"https://openalex.org/I57206974","display_name":"New York University","ror":"https://ror.org/0190ak572","country_code":"US","type":"education","lineage":["https://openalex.org/I57206974"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Margaret H. Wright","raw_affiliation_strings":["New York University"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"New York University","institution_ids":["https://openalex.org/I57206974"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":1,"citation_normalized_percentile":{"value":0.05688845,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"22","issue":"2","first_page":"501","last_page":"532"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.9937999844551086,"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":0.9937999844551086,"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":0.9212999939918518,"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/mathematics","display_name":"Mathematics","score":0.8356980085372925},{"id":"https://openalex.org/keywords/hessian-matrix","display_name":"Hessian matrix","score":0.5883923768997192},{"id":"https://openalex.org/keywords/bounded-function","display_name":"Bounded function","score":0.5548845529556274},{"id":"https://openalex.org/keywords/simplex-algorithm","display_name":"Simplex algorithm","score":0.5463564395904541},{"id":"https://openalex.org/keywords/simplex","display_name":"Simplex","score":0.5363588929176331},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.4730077087879181},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.4435917139053345},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.4281582236289978},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.416995108127594},{"id":"https://openalex.org/keywords/vertex","display_name":"Vertex (graph theory)","score":0.4131101667881012},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.31572526693344116},{"id":"https://openalex.org/keywords/linear-programming","display_name":"Linear programming","score":0.26598286628723145},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.20868796110153198},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.19023388624191284},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.10501289367675781},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.0997108519077301}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8356980085372925},{"id":"https://openalex.org/C203616005","wikidata":"https://www.wikidata.org/wiki/Q620495","display_name":"Hessian matrix","level":2,"score":0.5883923768997192},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.5548845529556274},{"id":"https://openalex.org/C144521790","wikidata":"https://www.wikidata.org/wiki/Q134164","display_name":"Simplex algorithm","level":3,"score":0.5463564395904541},{"id":"https://openalex.org/C62438384","wikidata":"https://www.wikidata.org/wiki/Q331350","display_name":"Simplex","level":2,"score":0.5363588929176331},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.4730077087879181},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.4435917139053345},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.4281582236289978},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.416995108127594},{"id":"https://openalex.org/C80899671","wikidata":"https://www.wikidata.org/wiki/Q1304193","display_name":"Vertex (graph theory)","level":3,"score":0.4131101667881012},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.31572526693344116},{"id":"https://openalex.org/C41045048","wikidata":"https://www.wikidata.org/wiki/Q202843","display_name":"Linear programming","level":2,"score":0.26598286628723145},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.20868796110153198},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.19023388624191284},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.10501289367675781},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.0997108519077301},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","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/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0}],"mesh":[],"locations_count":8,"locations":[{"id":"doi:10.1137/110830150","is_oa":false,"landing_page_url":"https://doi.org/10.1137/110830150","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:1104.0350","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1104.0350","pdf_url":"https://arxiv.org/pdf/1104.0350","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"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":null,"raw_type":"text"},{"id":"pmh:oai:dspace.mit.edu:1721.1/75022","is_oa":true,"landing_page_url":"http://hdl.handle.net/1721.1/75022","pdf_url":"https://dspace.mit.edu/bitstream/1721.1/75022/2/Lagarias-2012-Convergence%20of%20the%20Restricted%20Nelder--Mead%20Algorithm%20in%20Two%20Dimensions.pdf","source":{"id":"https://openalex.org/S4306400425","display_name":"DSpace@MIT (Massachusetts Institute of Technology)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I63966007","host_organization_name":"Massachusetts Institute of Technology","host_organization_lineage":["https://openalex.org/I63966007"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-nc","license_id":"https://openalex.org/licenses/cc-by-nc","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"SIAM","raw_type":"Article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.208.450","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.208.450","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www-math.mit.edu/%7Epoonen/papers/nm.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.227.4642","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.227.4642","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://cs.nyu.edu/faculty/mhw/rnm.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.765.9645","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.765.9645","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://arxiv.org/pdf/1104.0350.pdf","raw_type":"text"},{"id":"doi:10.48550/arxiv.1104.0350","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.1104.0350","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"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":"article-journal"},{"id":"mag:1597581062","is_oa":false,"landing_page_url":"https://dspace.mit.edu/handle/1721.1/75022","pdf_url":null,"source":{"id":"https://openalex.org/S897311980","display_name":"SIAM Journal on Control and Optimization","issn_l":"0363-0129","issn":["0363-0129","1095-7138"],"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":null,"is_accepted":false,"is_published":null,"raw_source_name":"SIAM Journal on Control and Optimization","raw_type":null}],"best_oa_location":{"id":"pmh:oai:arXiv.org:1104.0350","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1104.0350","pdf_url":"https://arxiv.org/pdf/1104.0350","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"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":null,"raw_type":"text"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G222768804","display_name":null,"funder_award_id":"DE\u2010FG02\u2010","funder_id":"https://openalex.org/F4320306084","funder_display_name":"U.S. Department of Energy"},{"id":"https://openalex.org/G3182508553","display_name":"Rational points on varieties in families, and countable unions of varieties over countable fields","funder_award_id":"0841321","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G3700750909","display_name":"ANALYTICAL AND COMPUTATIONAL  MATHEMATICAL PHYSICS","funder_award_id":"FG0288ER25053","funder_id":"https://openalex.org/F4320306084","funder_display_name":"U.S. Department of Energy"},{"id":"https://openalex.org/G8272312977","display_name":"APPLIED ANALYSIS AND COMPUTATIONAL MATHEMATICS","funder_award_id":"DEFG0288ER25053","funder_id":"https://openalex.org/F4320306084","funder_display_name":"U.S. Department of Energy"},{"id":"https://openalex.org/G8681799105","display_name":"Random maximal isotropic subspaces and Selmer groups","funder_award_id":"1069236","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G8963400735","display_name":"Eisenstein Series, Operators and L-Functions","funder_award_id":"0801029","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/F4320306084","display_name":"U.S. Department of Energy","ror":"https://ror.org/01bj3aw27"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":15,"referenced_works":["https://openalex.org/W1512858435","https://openalex.org/W1965748532","https://openalex.org/W1973518536","https://openalex.org/W1995832922","https://openalex.org/W1998855065","https://openalex.org/W2014337138","https://openalex.org/W2016387335","https://openalex.org/W2019742812","https://openalex.org/W2024991751","https://openalex.org/W2031316504","https://openalex.org/W2054875385","https://openalex.org/W2070922270","https://openalex.org/W2079777091","https://openalex.org/W2162737890","https://openalex.org/W2171074980"],"related_works":["https://openalex.org/W2950882985","https://openalex.org/W1565614474","https://openalex.org/W2742599086","https://openalex.org/W2018195621","https://openalex.org/W2404759370","https://openalex.org/W2893172655","https://openalex.org/W2116538711","https://openalex.org/W3151606440","https://openalex.org/W2058407505","https://openalex.org/W1971926417","https://openalex.org/W2963648037","https://openalex.org/W2155770401","https://openalex.org/W2272509765","https://openalex.org/W2198227614","https://openalex.org/W191473517","https://openalex.org/W2063196363","https://openalex.org/W1993553082","https://openalex.org/W3099872826","https://openalex.org/W2798832592","https://openalex.org/W2227972400"],"abstract_inverted_index":{"The":[0,173],"Nelder--Mead":[1,35,132],"algorithm,":[2,133],"a":[3,18,40,53,57,66,124,134,182],"longstanding":[4],"direct":[5],"search":[6],"method":[7,180],"for":[8,79,98,148,197],"unconstrained":[9,198],"optimization":[10],"published":[11],"in":[12,194],"1965,":[13],"is":[14,37,175],"designed":[15],"to":[16,95,110,123,170],"minimize":[17],"scalar-valued":[19],"function":[20,28,51,157],"$f$":[21],"of":[22,84,101,106],"$n$":[23],"real":[24],"variables":[25],"using":[26],"only":[27],"values,":[29],"without":[30],"any":[31,149,154],"derivative":[32],"information.":[33],"Each":[34],"iteration":[36,55],"associated":[38],"with":[39,87,158],"nondegenerate":[41,150],"simplex":[42,59,108,152],"defined":[43],"by":[44,60,65,115],"$n":[45],"+":[46],"1$":[47],"vertices":[48],"and":[49,153,162,186],"their":[50],"values;":[52],"typical":[54],"produces":[56],"new":[58,67],"replacing":[61],"the":[62,70,91,96,104,107,119,130,166,171,179],"worst":[63],"vertex":[64],"point.":[68,126],"Despite":[69],"method's":[71],"widespread":[72],"use,":[73],"theoretical":[74],"results":[75],"have":[76],"been":[77],"limited:":[78],"strictly":[80],"convex":[81],"objective":[82],"functions":[83,100],"one":[85],"variable":[86],"bounded":[88,163],"level":[89,164],"sets,":[90,165],"algorithm":[92,120,167],"always":[93,168],"converges":[94,109,169],"minimizer;":[97],"such":[99],"two":[102,143],"variables,":[103],"diameter":[105],"zero":[111],"but":[112],"examples":[113],"constructed":[114],"McKinnon":[116],"show":[117,146],"that":[118,136,147,191],"may":[121],"converge":[122],"nonminimizing":[125],"This":[127],"paper":[128],"considers":[129],"restricted":[131],"variant":[135],"does":[137],"not":[138],"allow":[139],"expansion":[140],"steps.":[141],"In":[142],"dimensions":[144],"we":[145],"starting":[151],"twice-continuously":[155],"differentiable":[156],"positive":[159],"definite":[160],"Hessian":[161],"minimizer.":[172],"proof":[174],"based":[176],"on":[177,188],"treating":[178],"as":[181],"discrete":[183],"dynamical":[184],"system":[185],"relies":[187],"several":[189],"techniques":[190],"are":[192],"nonstandard":[193],"convergence":[195],"proofs":[196],"optimization.":[199]},"counts_by_year":[{"year":2020,"cited_by_count":1}],"updated_date":"2026-08-18T07:49:30.821534","created_date":"2025-10-10T00:00:00"}