{"id":"https://openalex.org/W2598996488","doi":"https://doi.org/10.1137/16m1087333","title":"Weak, Strong, and Linear Convergence of a Double-Layer Fixed Point Algorithm","display_name":"Weak, Strong, and Linear Convergence of a Double-Layer Fixed Point Algorithm","publication_year":2017,"publication_date":"2017-01-01","ids":{"openalex":"https://openalex.org/W2598996488","doi":"https://doi.org/10.1137/16m1087333","mag":"2598996488"},"language":"en","primary_location":{"id":"doi:10.1137/16m1087333","is_oa":false,"landing_page_url":"https://doi.org/10.1137/16m1087333","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/A5001244633","display_name":"Victor I. Kolobov","orcid":null},"institutions":[{"id":"https://openalex.org/I174306211","display_name":"Technion \u2013 Israel Institute of Technology","ror":"https://ror.org/03qryx823","country_code":"IL","type":"education","lineage":["https://openalex.org/I174306211"]}],"countries":["IL"],"is_corresponding":false,"raw_author_name":"Victor I. Kolobov","raw_affiliation_strings":["Department of Computer Science, The Technion -Israel Institute of Technology, 32000 Haifa, Israel"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Computer Science, The Technion -Israel Institute of Technology, 32000 Haifa, Israel","institution_ids":["https://openalex.org/I174306211"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5057413218","display_name":"Simeon Reich","orcid":"https://orcid.org/0000-0003-0780-1559"},"institutions":[{"id":"https://openalex.org/I174306211","display_name":"Technion \u2013 Israel Institute of Technology","ror":"https://ror.org/03qryx823","country_code":"IL","type":"education","lineage":["https://openalex.org/I174306211"]}],"countries":["IL"],"is_corresponding":false,"raw_author_name":"Simeon Reich","raw_affiliation_strings":["Department of Mathematics, The Technion -Israel Institute of Technology, 32000 Haifa, Israel;"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, The Technion -Israel Institute of Technology, 32000 Haifa, Israel;","institution_ids":["https://openalex.org/I174306211"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5071059710","display_name":"Rafa\u0142 Zalas","orcid":"https://orcid.org/0000-0003-0193-9610"},"institutions":[{"id":"https://openalex.org/I174306211","display_name":"Technion \u2013 Israel Institute of Technology","ror":"https://ror.org/03qryx823","country_code":"IL","type":"education","lineage":["https://openalex.org/I174306211"]}],"countries":["IL"],"is_corresponding":false,"raw_author_name":"Rafa\u0142 Zalas","raw_affiliation_strings":["Department of Mathematics, The Technion -Israel Institute of Technology, 32000 Haifa, Israel;"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, The Technion -Israel Institute of Technology, 32000 Haifa, Israel;","institution_ids":["https://openalex.org/I174306211"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I174306211"],"apc_list":null,"apc_paid":null,"fwci":1.1961,"has_fulltext":false,"cited_by_count":14,"citation_normalized_percentile":{"value":0.80302438,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":97},"biblio":{"volume":"27","issue":"3","first_page":"1431","last_page":"1458"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10545","display_name":"Optimization and Variational Analysis","score":0.9998000264167786,"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/T10545","display_name":"Optimization and Variational Analysis","score":0.9998000264167786,"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/T11777","display_name":"Fixed Point Theorems Analysis","score":0.9932000041007996,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"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/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.9930999875068665,"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":0.8085606098175049},{"id":"https://openalex.org/keywords/subgradient-method","display_name":"Subgradient method","score":0.7116129398345947},{"id":"https://openalex.org/keywords/bounded-function","display_name":"Bounded function","score":0.6713935136795044},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.4927084743976593},{"id":"https://openalex.org/keywords/convex-function","display_name":"Convex function","score":0.473450630903244},{"id":"https://openalex.org/keywords/hilbert-space","display_name":"Hilbert space","score":0.46556949615478516},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.46048057079315186},{"id":"https://openalex.org/keywords/fixed-point","display_name":"Fixed point","score":0.4373522102832794},{"id":"https://openalex.org/keywords/projection","display_name":"Projection (relational algebra)","score":0.435366690158844},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.4296625852584839},{"id":"https://openalex.org/keywords/regular-polygon","display_name":"Regular polygon","score":0.4047022759914398},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.3965591788291931},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.37680989503860474},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.23052874207496643},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.21412038803100586},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.16287198662757874},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.08957767486572266}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8085606098175049},{"id":"https://openalex.org/C158968445","wikidata":"https://www.wikidata.org/wiki/Q7631150","display_name":"Subgradient method","level":2,"score":0.7116129398345947},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.6713935136795044},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.4927084743976593},{"id":"https://openalex.org/C145446738","wikidata":"https://www.wikidata.org/wiki/Q319913","display_name":"Convex function","level":3,"score":0.473450630903244},{"id":"https://openalex.org/C62799726","wikidata":"https://www.wikidata.org/wiki/Q190056","display_name":"Hilbert space","level":2,"score":0.46556949615478516},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.46048057079315186},{"id":"https://openalex.org/C61445026","wikidata":"https://www.wikidata.org/wiki/Q217608","display_name":"Fixed point","level":2,"score":0.4373522102832794},{"id":"https://openalex.org/C57493831","wikidata":"https://www.wikidata.org/wiki/Q3134666","display_name":"Projection (relational algebra)","level":2,"score":0.435366690158844},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4296625852584839},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.4047022759914398},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.3965591788291931},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.37680989503860474},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.23052874207496643},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.21412038803100586},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.16287198662757874},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.08957767486572266},{"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/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C50522688","wikidata":"https://www.wikidata.org/wiki/Q189833","display_name":"Economic growth","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/16m1087333","is_oa":false,"landing_page_url":"https://doi.org/10.1137/16m1087333","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/G2942734666","display_name":null,"funder_award_id":"389/12","funder_id":"https://openalex.org/F4320322252","funder_display_name":"Israel Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320322252","display_name":"Israel Science Foundation","ror":"https://ror.org/04sazxf24"},{"id":"https://openalex.org/F4320323721","display_name":"Technion-Israel Institute of Technology","ror":"https://ror.org/03qryx823"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":38,"referenced_works":["https://openalex.org/W1210354003","https://openalex.org/W1595803699","https://openalex.org/W1710475132","https://openalex.org/W1853237406","https://openalex.org/W1963892562","https://openalex.org/W1971915361","https://openalex.org/W1975181084","https://openalex.org/W1975552883","https://openalex.org/W1975567493","https://openalex.org/W1986280275","https://openalex.org/W1987059360","https://openalex.org/W1989820267","https://openalex.org/W2001032119","https://openalex.org/W2001504661","https://openalex.org/W2002110362","https://openalex.org/W2005693246","https://openalex.org/W2009126525","https://openalex.org/W2022993401","https://openalex.org/W2025424390","https://openalex.org/W2033511209","https://openalex.org/W2038085385","https://openalex.org/W2053487511","https://openalex.org/W2054796771","https://openalex.org/W2062059813","https://openalex.org/W2092527476","https://openalex.org/W2094508696","https://openalex.org/W2100885264","https://openalex.org/W2100904599","https://openalex.org/W2106454198","https://openalex.org/W2130469840","https://openalex.org/W2149059117","https://openalex.org/W2329579984","https://openalex.org/W2336954693","https://openalex.org/W2499924684","https://openalex.org/W2594309699","https://openalex.org/W2962704982","https://openalex.org/W3103639684","https://openalex.org/W4250507917"],"related_works":["https://openalex.org/W1991765628","https://openalex.org/W4362514784","https://openalex.org/W3164499765","https://openalex.org/W1551381264","https://openalex.org/W3005426141","https://openalex.org/W2491310570","https://openalex.org/W3126898700","https://openalex.org/W2591999657","https://openalex.org/W2015928709","https://openalex.org/W1983109191"],"abstract_inverted_index":{"In":[0,169,223],"this":[1,76,143,195],"article":[2],"we":[3,56,225,240],"consider":[4],"a":[5,11,17,45,52,88],"consistent":[6],"convex":[7],"feasibility":[8],"problem":[9],"in":[10,26,28,99,133,142,194,205],"real":[12],"Hilbert":[13],"space":[14],"defined":[15],"by":[16,125],"finite":[18],"family":[19],"of":[20,63,73,97,135,187],"sets":[21,121],"$C_i$.":[22,85],"We":[23,86],"are":[24],"interested,":[25],"particular,":[27,170],"the":[29,58,61,71,118,126,147,153,155,158,164,206,231,236],"case":[30],"where":[31,230],"for":[32,208],"each":[33],"$i$,":[34],"$C_i={Fix}":[35],"U_i=\\{z\\in":[36],"\\mathcal":[37,42],"H\\mid":[38],"p_i(z)=0\\}$,":[39],"$U_i\\colon\\mathcal":[40],"H\\rightarrow":[41,49],"H$":[43],"is":[44,51,65,77,131],"cutter":[46],"and":[47,75,157,166,177,184,220],"$p_i\\colon\\mathcal":[48],"[0,\\infty)$":[50],"proximity":[53,136,159],"function.":[54],"Moreover,":[55],"make":[57],"following":[59],"assumption:":[60],"computation":[62],"$p_i$":[64],"at":[66,78],"most":[67,79,119],"as":[68,70,80,82,239],"difficult":[69,81],"evaluation":[72],"$U_i$":[74],"projecting":[83],"onto":[84],"study":[87],"double-layer":[89,232],"fixed":[90],"point":[91],"algorithm":[92],"which":[93],"applies":[94],"two":[95],"types":[96],"controls":[98],"every":[100],"iteration":[101],"step.":[102],"The":[103,113,129,138,191],"first":[104,127],"one---the":[105,115],"outer":[106,167],"control---is":[107],"assumed":[108],"to":[109],"be":[110],"almost":[111],"cyclic.":[112],"second":[114],"inner":[116,165],"control---determines":[117],"important":[120],"from":[122],"those":[123,210],"offered":[124],"one.":[128],"selection":[130],"made":[132],"terms":[134],"functions.":[137],"convergence":[139,186,237],"results":[140],"presented":[141,193],"manuscript":[144],"depend":[145],"on":[146],"conditions":[148],"which,":[149],"first,":[150],"bind":[151],"together":[152],"sets,":[154],"operators,":[156],"functions":[160],"and,":[161],"second,":[162],"connect":[163],"controls.":[168,222],"weak":[171],"regularity":[172,180],"(demi-closedness":[173],"principle),":[174],"bounded":[175,178],"regularity,":[176],"linear":[179,185],"imply":[181],"weak,":[182],"strong,":[183],"our":[188],"algorithm,":[189],"respectively.":[190],"framework":[192],"paper":[196],"covers":[197],"many":[198],"known":[199],"(subgradient)":[200],"projection":[201],"algorithms":[202],"already":[203],"existing":[204],"literature,":[207],"example,":[209],"applied":[211],"with":[212],"(almost)":[213],"cyclic,":[214],"remotest-set,":[215],"maximum":[216],"displacement,":[217],"most-violated":[218],"constraint,":[219],"simultaneous":[221],"addition,":[224],"provide":[226],"several":[227],"new":[228],"examples,":[229],"approach":[233],"indeed":[234],"accelerates":[235],"speed":[238],"demonstrate":[241],"numerically.":[242]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":2},{"year":2021,"cited_by_count":3},{"year":2020,"cited_by_count":1},{"year":2018,"cited_by_count":2},{"year":2017,"cited_by_count":2}],"updated_date":"2026-06-26T08:34:08.712188","created_date":"2025-10-10T00:00:00"}