{"id":"https://openalex.org/W2985739199","doi":"https://doi.org/10.1137/18m1167061","title":"Randomized Projection Methods for Convex Feasibility: Conditioning and Convergence Rates","display_name":"Randomized Projection Methods for Convex Feasibility: Conditioning and Convergence Rates","publication_year":2019,"publication_date":"2019-01-01","ids":{"openalex":"https://openalex.org/W2985739199","doi":"https://doi.org/10.1137/18m1167061","mag":"2985739199"},"language":"en","primary_location":{"id":"doi:10.1137/18m1167061","is_oa":false,"landing_page_url":"https://doi.org/10.1137/18m1167061","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/A5019114471","display_name":"Ion Necoara","orcid":"https://orcid.org/0000-0003-1102-2654"},"institutions":[{"id":"https://openalex.org/I61641377","display_name":"Universitatea Na\u021bional\u0103 de \u0218tiin\u021b\u0103 \u0219i Tehnologie Politehnica Bucure\u0219ti","ror":"https://ror.org/0558j5q12","country_code":"RO","type":"education","lineage":["https://openalex.org/I61641377"]}],"countries":["RO"],"is_corresponding":true,"raw_author_name":"Ion Necoara","raw_affiliation_strings":["Automatic Control and Systems Engineering Department, University Politehnica Bucharest, 060042 Bucharest, Romania"],"affiliations":[{"raw_affiliation_string":"Automatic Control and Systems Engineering Department, University Politehnica Bucharest, 060042 Bucharest, Romania","institution_ids":["https://openalex.org/I61641377"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5036598221","display_name":"Peter Richt\u00e1rik","orcid":"https://orcid.org/0000-0003-4380-5848"},"institutions":[{"id":"https://openalex.org/I61641377","display_name":"Universitatea Na\u021bional\u0103 de \u0218tiin\u021b\u0103 \u0219i Tehnologie Politehnica Bucure\u0219ti","ror":"https://ror.org/0558j5q12","country_code":"RO","type":"education","lineage":["https://openalex.org/I61641377"]}],"countries":["RO"],"is_corresponding":false,"raw_author_name":"Peter Richt\u00e1rik","raw_affiliation_strings":["Automatic Control and Systems Engineering Department, University Politehnica Bucharest, 060042 Bucharest, Romania"],"affiliations":[{"raw_affiliation_string":"Automatic Control and Systems Engineering Department, University Politehnica Bucharest, 060042 Bucharest, Romania","institution_ids":["https://openalex.org/I61641377"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5049988389","display_name":"Andrei P\u0103tra\u015fcu","orcid":"https://orcid.org/0000-0002-9293-9386"},"institutions":[{"id":"https://openalex.org/I61641377","display_name":"Universitatea Na\u021bional\u0103 de \u0218tiin\u021b\u0103 \u0219i Tehnologie Politehnica Bucure\u0219ti","ror":"https://ror.org/0558j5q12","country_code":"RO","type":"education","lineage":["https://openalex.org/I61641377"]}],"countries":["RO"],"is_corresponding":false,"raw_author_name":"Andrei Patrascu","raw_affiliation_strings":["Automatic Control and Systems Engineering Department, University Politehnica Bucharest, 060042 Bucharest, Romania"],"affiliations":[{"raw_affiliation_string":"Automatic Control and Systems Engineering Department, University Politehnica Bucharest, 060042 Bucharest, Romania","institution_ids":["https://openalex.org/I61641377"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5019114471"],"corresponding_institution_ids":["https://openalex.org/I61641377"],"apc_list":null,"apc_paid":null,"fwci":6.2108,"has_fulltext":false,"cited_by_count":42,"citation_normalized_percentile":{"value":0.97898563,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":99},"biblio":{"volume":"29","issue":"4","first_page":"2814","last_page":"2852"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.9998000264167786,"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"}},"topics":[{"id":"https://openalex.org/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.9998000264167786,"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/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.9994999766349792,"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"}},{"id":"https://openalex.org/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.9962000250816345,"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":0.7764222025871277},{"id":"https://openalex.org/keywords/stochastic-optimization","display_name":"Stochastic optimization","score":0.6502653360366821},{"id":"https://openalex.org/keywords/projection","display_name":"Projection (relational algebra)","score":0.5742793679237366},{"id":"https://openalex.org/keywords/lipschitz-continuity","display_name":"Lipschitz continuity","score":0.5703206062316895},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.5628859400749207},{"id":"https://openalex.org/keywords/convex-optimization","display_name":"Convex optimization","score":0.5326380729675293},{"id":"https://openalex.org/keywords/subgradient-method","display_name":"Subgradient method","score":0.5170615315437317},{"id":"https://openalex.org/keywords/constant","display_name":"Constant (computer programming)","score":0.48426076769828796},{"id":"https://openalex.org/keywords/independent-and-identically-distributed-random-variables","display_name":"Independent and identically distributed random variables","score":0.46372315287590027},{"id":"https://openalex.org/keywords/convex-set","display_name":"Convex set","score":0.4559295177459717},{"id":"https://openalex.org/keywords/optimization-problem","display_name":"Optimization problem","score":0.44968077540397644},{"id":"https://openalex.org/keywords/stochastic-approximation","display_name":"Stochastic approximation","score":0.4450918734073639},{"id":"https://openalex.org/keywords/regular-polygon","display_name":"Regular polygon","score":0.425912082195282},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.41344547271728516},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4050198197364807},{"id":"https://openalex.org/keywords/random-variable","display_name":"Random variable","score":0.2994617819786072},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.19350776076316833},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.170772522687912},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.1318562626838684},{"id":"https://openalex.org/keywords/key","display_name":"Key (lock)","score":0.09634828567504883},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.08341595530509949}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7764222025871277},{"id":"https://openalex.org/C194387892","wikidata":"https://www.wikidata.org/wiki/Q1747770","display_name":"Stochastic optimization","level":2,"score":0.6502653360366821},{"id":"https://openalex.org/C57493831","wikidata":"https://www.wikidata.org/wiki/Q3134666","display_name":"Projection (relational algebra)","level":2,"score":0.5742793679237366},{"id":"https://openalex.org/C22324862","wikidata":"https://www.wikidata.org/wiki/Q652707","display_name":"Lipschitz continuity","level":2,"score":0.5703206062316895},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.5628859400749207},{"id":"https://openalex.org/C157972887","wikidata":"https://www.wikidata.org/wiki/Q463359","display_name":"Convex optimization","level":3,"score":0.5326380729675293},{"id":"https://openalex.org/C158968445","wikidata":"https://www.wikidata.org/wiki/Q7631150","display_name":"Subgradient method","level":2,"score":0.5170615315437317},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.48426076769828796},{"id":"https://openalex.org/C141513077","wikidata":"https://www.wikidata.org/wiki/Q378542","display_name":"Independent and identically distributed random variables","level":3,"score":0.46372315287590027},{"id":"https://openalex.org/C49870271","wikidata":"https://www.wikidata.org/wiki/Q193657","display_name":"Convex set","level":4,"score":0.4559295177459717},{"id":"https://openalex.org/C137836250","wikidata":"https://www.wikidata.org/wiki/Q984063","display_name":"Optimization problem","level":2,"score":0.44968077540397644},{"id":"https://openalex.org/C55479107","wikidata":"https://www.wikidata.org/wiki/Q97663916","display_name":"Stochastic approximation","level":3,"score":0.4450918734073639},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.425912082195282},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.41344547271728516},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4050198197364807},{"id":"https://openalex.org/C122123141","wikidata":"https://www.wikidata.org/wiki/Q176623","display_name":"Random variable","level":2,"score":0.2994617819786072},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.19350776076316833},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.170772522687912},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.1318562626838684},{"id":"https://openalex.org/C26517878","wikidata":"https://www.wikidata.org/wiki/Q228039","display_name":"Key (lock)","level":2,"score":0.09634828567504883},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.08341595530509949},{"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/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","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/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","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}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1137/18m1167061","is_oa":false,"landing_page_url":"https://doi.org/10.1137/18m1167061","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:repository.kaust.edu.sa:10754/660992","is_oa":false,"landing_page_url":"http://hdl.handle.net/10754/660992","pdf_url":null,"source":{"id":"https://openalex.org/S4306401596","display_name":"King Abdullah University of Science and Technology Repository (King Abdullah University of Science and Technology)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I71920554","host_organization_name":"King Abdullah University of Science and Technology","host_organization_lineage":["https://openalex.org/I71920554"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G5973857375","display_name":null,"funder_award_id":"PNIII-P4-PCE-2016-0731","funder_id":"https://openalex.org/F4320323983","funder_display_name":"Unitatea Executiva pentru Finantarea Invatamantului Superior, a Cercetarii, Dezvoltarii si Inovarii"}],"funders":[{"id":"https://openalex.org/F4320323983","display_name":"Unitatea Executiva pentru Finantarea Invatamantului Superior, a Cercetarii, Dezvoltarii si Inovarii","ror":"https://ror.org/01q7jq182"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":23,"referenced_works":["https://openalex.org/W292990780","https://openalex.org/W1964744380","https://openalex.org/W1975552883","https://openalex.org/W2012180727","https://openalex.org/W2030161963","https://openalex.org/W2033511209","https://openalex.org/W2048305372","https://openalex.org/W2064114733","https://openalex.org/W2066333552","https://openalex.org/W2067877923","https://openalex.org/W2073749939","https://openalex.org/W2081633016","https://openalex.org/W2085440994","https://openalex.org/W2100885264","https://openalex.org/W2113503719","https://openalex.org/W2131247345","https://openalex.org/W2153374208","https://openalex.org/W2162461247","https://openalex.org/W2167820643","https://openalex.org/W2321466613","https://openalex.org/W2963830980","https://openalex.org/W3099862346","https://openalex.org/W4244061773"],"related_works":["https://openalex.org/W1556217901","https://openalex.org/W2012437081","https://openalex.org/W4221167213","https://openalex.org/W2920841750","https://openalex.org/W125581526","https://openalex.org/W3188394678","https://openalex.org/W2808103846","https://openalex.org/W2124181538","https://openalex.org/W3118796771","https://openalex.org/W2784222105"],"abstract_inverted_index":{"In":[0,43,84,167,249],"this":[1,85],"paper":[2],"we":[3,45,93,170,251],"develop":[4],"a":[5,72,77,95,125,141,145,149,201,214,275],"family":[6],"of":[7,31,38,97,109,152,164,174,204,207,222,240,244,278],"randomized":[8,223],"projection":[9,224,247],"methods":[10],"(RPM)":[11],"for":[12,71,217],"solving":[13],"the":[14,39,57,63,110,153,162,172,187,205,208,218,233,237,259,262],"convex":[15,32,40,179],"feasibility":[16,41,65,180],"problem.":[17,42,66],"Our":[18],"approach":[19],"is":[20,267],"based":[21],"on":[22,113,161,186],"several":[23],"new":[24],"ideas":[25],"and":[26,36,53,62,103,117,157,191,220],"tools,":[27],"including":[28],"stochastic":[29,34,49,51,58,64,134,154,209,263],"approximation":[30],"sets,":[33,116],"reformulations,":[35],"conditioning":[37],"particular,":[44],"propose":[46],"four":[47],"equivalent":[48],"reformulations:":[50],"smooth":[52],"nonsmooth":[54],"optimization":[55,155,210],"problems,":[56],"fixed":[59],"point":[60,73],"problem,":[61],"The":[67],"last":[68],"reformulation":[69,156],"asks":[70],"which":[74,196],"belongs":[75],"to":[76,177,236],"certain":[78],"random":[79,98],"set":[80],"with":[81,124,140],"probability":[82],"one.":[83],"case,":[86],"RPM":[87,137],"can":[88,197],"be":[89,198],"interpreted":[90,199],"as":[91,148,200],"follows:":[92],"sample":[94],"batch":[96],"sets":[99,165],"in":[100,232,258],"an":[101,192,229],"independently":[102],"identically":[104],"distributed":[105],"fashion,":[106],"perform":[107],"projections":[108,121],"current":[111],"iterate":[112],"all":[114],"sampled":[115],"then":[118],"combine":[119],"these":[120],"by":[122],"averaging":[123],"carefully":[126],"designed":[127],"extrapolation":[128],"step.":[129],"We":[130],"prove":[131,252],"that":[132,143,158,253],"under":[133],"linear":[135,188,264],"regularity,":[136],"converges":[138,256],"linearly,":[139],"rate":[142],"has":[144],"natural":[146],"interpretation":[147],"condition":[150,175,183,266],"number":[151,163,176,184],"depends":[159,185],"explicitly":[160],"sampled.":[166],"doing":[168],"so,":[169],"extend":[171],"concept":[173],"general":[178,215],"problems.":[181],"This":[182],"regularity":[189,265],"constant":[190,195,203,285],"additional":[193],"key":[194],"Lipschitz":[202],"gradient":[206],"reformulation.":[211],"Besides":[212],"providing":[213],"framework":[216],"design":[219],"analysis":[221],"schemes,":[225],"our":[226,254,279],"results":[227,272],"resolve":[228],"open":[230],"problem":[231],"literature":[234],"related":[235],"theoretical":[238],"understanding":[239],"observed":[241],"practical":[242],"efficiency":[243],"extrapolated":[245,280],"parallel":[246],"methods.":[248],"addition,":[250],"method":[255],"sublinearly":[257],"case":[260],"when":[261],"not":[268],"satisfied.":[269],"Preliminary":[270],"numerical":[271],"also":[273],"show":[274],"better":[276],"performance":[277],"step-size":[281,286],"scheme":[282],"over":[283],"its":[284],"counterpart.":[287]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":4},{"year":2024,"cited_by_count":4},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":7},{"year":2021,"cited_by_count":9},{"year":2020,"cited_by_count":10},{"year":2019,"cited_by_count":6}],"updated_date":"2026-04-14T08:04:32.555800","created_date":"2025-10-10T00:00:00"}