{"id":"https://openalex.org/W4388217958","doi":"https://doi.org/10.1007/s10957-023-02320-2","title":"Bregman-Golden Ratio Algorithms for Variational Inequalities","display_name":"Bregman-Golden Ratio Algorithms for Variational Inequalities","publication_year":2023,"publication_date":"2023-11-02","ids":{"openalex":"https://openalex.org/W4388217958","doi":"https://doi.org/10.1007/s10957-023-02320-2"},"language":"en","primary_location":{"id":"doi:10.1007/s10957-023-02320-2","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10957-023-02320-2","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10957-023-02320-2.pdf","source":{"id":"https://openalex.org/S52430896","display_name":"Journal of Optimization Theory and Applications","issn_l":"0022-3239","issn":["0022-3239","1573-2878"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Optimization Theory and Applications","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s10957-023-02320-2.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5029600817","display_name":"Matthew K. Tam","orcid":"https://orcid.org/0000-0002-3654-6553"},"institutions":[{"id":"https://openalex.org/I165779595","display_name":"The University of Melbourne","ror":"https://ror.org/01ej9dk98","country_code":"AU","type":"education","lineage":["https://openalex.org/I165779595"]}],"countries":["AU"],"is_corresponding":true,"raw_author_name":"Matthew K. Tam","raw_affiliation_strings":["School of Mathematics and Statistics, The University of Melbourne, Parkville, VIC, 3010, Australia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematics and Statistics, The University of Melbourne, Parkville, VIC, 3010, Australia","institution_ids":["https://openalex.org/I165779595"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5015278986","display_name":"Daniel J. Uteda","orcid":"https://orcid.org/0000-0003-4166-3452"},"institutions":[{"id":"https://openalex.org/I165779595","display_name":"The University of Melbourne","ror":"https://ror.org/01ej9dk98","country_code":"AU","type":"education","lineage":["https://openalex.org/I165779595"]}],"countries":["AU"],"is_corresponding":false,"raw_author_name":"Daniel J. Uteda","raw_affiliation_strings":["School of Mathematics and Statistics, The University of Melbourne, Parkville, VIC, 3010, Australia"],"raw_orcid":"https://orcid.org/0000-0003-4166-3452","affiliations":[{"raw_affiliation_string":"School of Mathematics and Statistics, The University of Melbourne, Parkville, VIC, 3010, Australia","institution_ids":["https://openalex.org/I165779595"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5029600817"],"corresponding_institution_ids":["https://openalex.org/I165779595"],"apc_list":{"value":2390,"currency":"EUR","value_usd":2990},"apc_paid":{"value":2390,"currency":"EUR","value_usd":2990},"fwci":2.4712,"has_fulltext":true,"cited_by_count":13,"citation_normalized_percentile":{"value":0.91127554,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":97,"max":100},"biblio":{"volume":"199","issue":"3","first_page":"993","last_page":"1021"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10545","display_name":"Optimization and Variational Analysis","score":0.9994999766349792,"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":0.9994999766349792,"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":0.9922000169754028,"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":0.9830999970436096,"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/theory-of-computation","display_name":"Theory of computation","score":0.793705940246582},{"id":"https://openalex.org/keywords/variational-inequality","display_name":"Variational inequality","score":0.7312443256378174},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7273653745651245},{"id":"https://openalex.org/keywords/bregman-divergence","display_name":"Bregman divergence","score":0.691476583480835},{"id":"https://openalex.org/keywords/saddle-point","display_name":"Saddle point","score":0.638329267501831},{"id":"https://openalex.org/keywords/lipschitz-continuity","display_name":"Lipschitz continuity","score":0.6144904494285583},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.5229516625404358},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.5087898373603821},{"id":"https://openalex.org/keywords/benchmark","display_name":"Benchmark (surveying)","score":0.5041805505752563},{"id":"https://openalex.org/keywords/subgradient-method","display_name":"Subgradient method","score":0.43750664591789246},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.37017494440078735},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.33005860447883606},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.10530561208724976},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.0748445987701416}],"concepts":[{"id":"https://openalex.org/C24858836","wikidata":"https://www.wikidata.org/wiki/Q844718","display_name":"Theory of computation","level":2,"score":0.793705940246582},{"id":"https://openalex.org/C161999928","wikidata":"https://www.wikidata.org/wiki/Q4556320","display_name":"Variational inequality","level":2,"score":0.7312443256378174},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7273653745651245},{"id":"https://openalex.org/C149073432","wikidata":"https://www.wikidata.org/wiki/Q4960382","display_name":"Bregman divergence","level":2,"score":0.691476583480835},{"id":"https://openalex.org/C2681867","wikidata":"https://www.wikidata.org/wiki/Q690935","display_name":"Saddle point","level":2,"score":0.638329267501831},{"id":"https://openalex.org/C22324862","wikidata":"https://www.wikidata.org/wiki/Q652707","display_name":"Lipschitz continuity","level":2,"score":0.6144904494285583},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.5229516625404358},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.5087898373603821},{"id":"https://openalex.org/C185798385","wikidata":"https://www.wikidata.org/wiki/Q1161707","display_name":"Benchmark (surveying)","level":2,"score":0.5041805505752563},{"id":"https://openalex.org/C158968445","wikidata":"https://www.wikidata.org/wiki/Q7631150","display_name":"Subgradient method","level":2,"score":0.43750664591789246},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.37017494440078735},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.33005860447883606},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.10530561208724976},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0748445987701416},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","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},{"id":"https://openalex.org/C205649164","wikidata":"https://www.wikidata.org/wiki/Q1071","display_name":"Geography","level":0,"score":0.0},{"id":"https://openalex.org/C13280743","wikidata":"https://www.wikidata.org/wiki/Q131089","display_name":"Geodesy","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1007/s10957-023-02320-2","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10957-023-02320-2","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10957-023-02320-2.pdf","source":{"id":"https://openalex.org/S52430896","display_name":"Journal of Optimization Theory and Applications","issn_l":"0022-3239","issn":["0022-3239","1573-2878"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Optimization Theory and Applications","raw_type":"journal-article"},{"id":"pmh:oai:RePEc:spr:joptap:v:199:y:2023:i:3:d:10.1007_s10957-023-02320-2","is_oa":false,"landing_page_url":"http://link.springer.com/10.1007/s10957-023-02320-2","pdf_url":null,"source":{"id":"https://openalex.org/S4306401271","display_name":"RePEc: Research Papers in Economics","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I77793887","host_organization_name":"Federal Reserve Bank of St. Louis","host_organization_lineage":["https://openalex.org/I77793887"],"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":"article"}],"best_oa_location":{"id":"doi:10.1007/s10957-023-02320-2","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10957-023-02320-2","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10957-023-02320-2.pdf","source":{"id":"https://openalex.org/S52430896","display_name":"Journal of Optimization Theory and Applications","issn_l":"0022-3239","issn":["0022-3239","1573-2878"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Optimization Theory and Applications","raw_type":"journal-article"},"sustainable_development_goals":[{"display_name":"Reduced inequalities","id":"https://metadata.un.org/sdg/10","score":0.6200000047683716}],"awards":[{"id":"https://openalex.org/G3948230744","display_name":null,"funder_award_id":"DE200100063","funder_id":"https://openalex.org/F4320334704","funder_display_name":"Australian Research Council"}],"funders":[{"id":"https://openalex.org/F4320334704","display_name":"Australian Research Council","ror":"https://ror.org/05mmh0f86"}],"has_content":{"grobid_xml":false,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4388217958.pdf"},"referenced_works_count":41,"referenced_works":["https://openalex.org/W1583837637","https://openalex.org/W1981276685","https://openalex.org/W1981558178","https://openalex.org/W1985348591","https://openalex.org/W2016384870","https://openalex.org/W2022508825","https://openalex.org/W2023736708","https://openalex.org/W2027711030","https://openalex.org/W2030392655","https://openalex.org/W2057264006","https://openalex.org/W2081316950","https://openalex.org/W2089725613","https://openalex.org/W2090376964","https://openalex.org/W2094548886","https://openalex.org/W2150867680","https://openalex.org/W2478708596","https://openalex.org/W2516732819","https://openalex.org/W2551153860","https://openalex.org/W2800693632","https://openalex.org/W2926523766","https://openalex.org/W2945578891","https://openalex.org/W2950866122","https://openalex.org/W2965900084","https://openalex.org/W3003500322","https://openalex.org/W3023751834","https://openalex.org/W3027891624","https://openalex.org/W3046538318","https://openalex.org/W3095801303","https://openalex.org/W3095937564","https://openalex.org/W3101497372","https://openalex.org/W3106943118","https://openalex.org/W3144865899","https://openalex.org/W3161273445","https://openalex.org/W3164046124","https://openalex.org/W3217517582","https://openalex.org/W4226197130","https://openalex.org/W4229650096","https://openalex.org/W4303084602","https://openalex.org/W6600002382","https://openalex.org/W6600339457","https://openalex.org/W6605035644"],"related_works":["https://openalex.org/W2078980565","https://openalex.org/W2962762987","https://openalex.org/W2781777036","https://openalex.org/W3196983597","https://openalex.org/W2029580202","https://openalex.org/W2755725405","https://openalex.org/W2503607608","https://openalex.org/W1991765628","https://openalex.org/W4206677478","https://openalex.org/W2923595680"],"abstract_inverted_index":{"Abstract":[0],"Variational":[1],"inequalities":[2],"provide":[3],"a":[4,37,65,92],"framework":[5],"through":[6],"which":[7,35],"many":[8],"optimisation":[9],"problems":[10],"can":[11],"be":[12],"solved,":[13],"in":[14,96,109],"particular,":[15],"saddle-point":[16],"problems.":[17],"In":[18,119],"this":[19],"paper,":[20],"we":[21],"study":[22],"modifications":[23,59],"to":[24,60],"the":[25,62,76,86,101,130,137],"so-called":[26],"Golden":[27],"RAtio":[28],"ALgorithm":[29],"(GRAAL)":[30],"for":[31],"variational":[32],"inequalities\u2014a":[33],"method":[34,88],"uses":[36,64,78],"fully":[38],"explicit":[39],"adaptive":[40,80],"step-size":[41,81],"and":[42,55,69,75,99,113],"provides":[43],"convergence":[44],"results":[45],"under":[46,71],"local":[47],"Lipschitz":[48,73],"assumptions":[49],"without":[50],"requiring":[51],"backtracking.":[52],"We":[53],"present":[54],"analyse":[56],"two":[57,104],"Bregman":[58,127],"GRAAL:":[61],"first":[63],"fixed":[66],"step":[67],"size":[68],"converges":[70],"global":[72],"assumptions,":[74],"second":[77],"an":[79,124],"rule.":[82],"Numerical":[83],"performance":[84],"of":[85,100,121,136],"former":[87],"is":[89],"demonstrated":[90],"on":[91,103],"bimatrix":[93],"game":[94],"arising":[95],"network":[97],"communication,":[98],"latter":[102],"problems,":[105],"namely,":[106],"power":[107],"allocation":[108],"Gaussian":[110],"communication":[111],"channels":[112],"N":[114],"-person":[115],"Cournot":[116],"completion":[117],"games.":[118],"all":[120],"these":[122],"applications,":[123],"appropriately":[125],"chosen":[126],"distance":[128],"simplifies":[129],"projection":[131],"steps":[132],"computed":[133],"as":[134],"part":[135],"algorithm.":[138]},"counts_by_year":[{"year":2026,"cited_by_count":5},{"year":2025,"cited_by_count":4},{"year":2024,"cited_by_count":4}],"updated_date":"2026-05-21T09:19:25.381259","created_date":"2025-10-10T00:00:00"}
