{"id":"https://openalex.org/W2149859242","doi":"https://doi.org/10.1109/tit.2011.2132470","title":"A Generalization of Nash Bargaining and Proportional Fairness to Log-Convex Utility Sets With Power Constraints","display_name":"A Generalization of Nash Bargaining and Proportional Fairness to Log-Convex Utility Sets With Power Constraints","publication_year":2011,"publication_date":"2011-05-25","ids":{"openalex":"https://openalex.org/W2149859242","doi":"https://doi.org/10.1109/tit.2011.2132470","mag":"2149859242"},"language":"en","primary_location":{"id":"doi:10.1109/tit.2011.2132470","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.2011.2132470","pdf_url":null,"source":{"id":"https://openalex.org/S4502562","display_name":"IEEE Transactions on Information Theory","issn_l":"0018-9448","issn":["0018-9448","1557-9654"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Information Theory","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"http://mediatum.ub.tum.de/node?id=1081677","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5000732219","display_name":"Holger Boche","orcid":"https://orcid.org/0000-0002-8375-8946"},"institutions":[{"id":"https://openalex.org/I62916508","display_name":"Technical University of Munich","ror":"https://ror.org/02kkvpp62","country_code":"DE","type":"education","lineage":["https://openalex.org/I62916508"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Holger Boche","raw_affiliation_strings":["Fraunhofer Institute for Telecommunications, Technical University Berlin, Germany","Technische Universit\u00e4t M\u00fcnchen, Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Fraunhofer Institute for Telecommunications, Technical University Berlin, Germany","institution_ids":[]},{"raw_affiliation_string":"Technische Universit\u00e4t M\u00fcnchen, Germany","institution_ids":["https://openalex.org/I62916508"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5038859625","display_name":"Martin Schubert","orcid":"https://orcid.org/0000-0003-4495-6533"},"institutions":[{"id":"https://openalex.org/I2800274787","display_name":"Fraunhofer Institute for Telecommunications, Heinrich Hertz Institute","ror":"https://ror.org/02tbr6331","country_code":"DE","type":"facility","lineage":["https://openalex.org/I2800274787","https://openalex.org/I4923324"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Martin Schubert","raw_affiliation_strings":["Fraunhofer Institute for Telecommunications, HHI, Berlin, Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Fraunhofer Institute for Telecommunications, HHI, Berlin, Germany","institution_ids":["https://openalex.org/I2800274787"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":2.3964,"has_fulltext":false,"cited_by_count":17,"citation_normalized_percentile":{"value":0.89795741,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":99},"biblio":{"volume":"57","issue":"6","first_page":"3390","last_page":"3404"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10148","display_name":"Advanced MIMO Systems Optimization","score":0.9955999851226807,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"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/T10148","display_name":"Advanced MIMO Systems Optimization","score":0.9955999851226807,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"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/T11409","display_name":"Advanced Wireless Network Optimization","score":0.9944999814033508,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"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/T10796","display_name":"Cooperative Communication and Network Coding","score":0.9925000071525574,"subfield":{"id":"https://openalex.org/subfields/1705","display_name":"Computer Networks and Communications"},"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/convexity","display_name":"Convexity","score":0.7259837985038757},{"id":"https://openalex.org/keywords/nash-equilibrium","display_name":"Nash equilibrium","score":0.5913407206535339},{"id":"https://openalex.org/keywords/bargaining-problem","display_name":"Bargaining problem","score":0.5776209831237793},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.5703415870666504},{"id":"https://openalex.org/keywords/generalization","display_name":"Generalization","score":0.5456836223602295},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5327563285827637},{"id":"https://openalex.org/keywords/convex-set","display_name":"Convex set","score":0.525416374206543},{"id":"https://openalex.org/keywords/constraint","display_name":"Constraint (computer-aided design)","score":0.4794839322566986},{"id":"https://openalex.org/keywords/subderivative","display_name":"Subderivative","score":0.46325716376304626},{"id":"https://openalex.org/keywords/boundary","display_name":"Boundary (topology)","score":0.46191859245300293},{"id":"https://openalex.org/keywords/mathematical-economics","display_name":"Mathematical economics","score":0.4522148668766022},{"id":"https://openalex.org/keywords/logarithm","display_name":"Logarithm","score":0.4483644664287567},{"id":"https://openalex.org/keywords/game-theory","display_name":"Game theory","score":0.42891374230384827},{"id":"https://openalex.org/keywords/convex-analysis","display_name":"Convex analysis","score":0.42790257930755615},{"id":"https://openalex.org/keywords/pareto-principle","display_name":"Pareto principle","score":0.41416096687316895},{"id":"https://openalex.org/keywords/convex-function","display_name":"Convex function","score":0.4110298454761505},{"id":"https://openalex.org/keywords/regular-polygon","display_name":"Regular polygon","score":0.4100888967514038},{"id":"https://openalex.org/keywords/convex-optimization","display_name":"Convex optimization","score":0.3141734302043915},{"id":"https://openalex.org/keywords/economics","display_name":"Economics","score":0.15066730976104736}],"concepts":[{"id":"https://openalex.org/C72134830","wikidata":"https://www.wikidata.org/wiki/Q5166524","display_name":"Convexity","level":2,"score":0.7259837985038757},{"id":"https://openalex.org/C46814582","wikidata":"https://www.wikidata.org/wiki/Q23389","display_name":"Nash equilibrium","level":2,"score":0.5913407206535339},{"id":"https://openalex.org/C90376892","wikidata":"https://www.wikidata.org/wiki/Q1516247","display_name":"Bargaining problem","level":2,"score":0.5776209831237793},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.5703415870666504},{"id":"https://openalex.org/C177148314","wikidata":"https://www.wikidata.org/wiki/Q170084","display_name":"Generalization","level":2,"score":0.5456836223602295},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5327563285827637},{"id":"https://openalex.org/C49870271","wikidata":"https://www.wikidata.org/wiki/Q193657","display_name":"Convex set","level":4,"score":0.525416374206543},{"id":"https://openalex.org/C2776036281","wikidata":"https://www.wikidata.org/wiki/Q48769818","display_name":"Constraint (computer-aided design)","level":2,"score":0.4794839322566986},{"id":"https://openalex.org/C200661725","wikidata":"https://www.wikidata.org/wiki/Q289500","display_name":"Subderivative","level":4,"score":0.46325716376304626},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.46191859245300293},{"id":"https://openalex.org/C144237770","wikidata":"https://www.wikidata.org/wiki/Q747534","display_name":"Mathematical economics","level":1,"score":0.4522148668766022},{"id":"https://openalex.org/C39927690","wikidata":"https://www.wikidata.org/wiki/Q11197","display_name":"Logarithm","level":2,"score":0.4483644664287567},{"id":"https://openalex.org/C177142836","wikidata":"https://www.wikidata.org/wiki/Q44455","display_name":"Game theory","level":2,"score":0.42891374230384827},{"id":"https://openalex.org/C12108790","wikidata":"https://www.wikidata.org/wiki/Q2234833","display_name":"Convex analysis","level":4,"score":0.42790257930755615},{"id":"https://openalex.org/C137635306","wikidata":"https://www.wikidata.org/wiki/Q182667","display_name":"Pareto principle","level":2,"score":0.41416096687316895},{"id":"https://openalex.org/C145446738","wikidata":"https://www.wikidata.org/wiki/Q319913","display_name":"Convex function","level":3,"score":0.4110298454761505},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.4100888967514038},{"id":"https://openalex.org/C157972887","wikidata":"https://www.wikidata.org/wiki/Q463359","display_name":"Convex optimization","level":3,"score":0.3141734302043915},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.15066730976104736},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0},{"id":"https://openalex.org/C106159729","wikidata":"https://www.wikidata.org/wiki/Q2294553","display_name":"Financial economics","level":1,"score":0.0},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1109/tit.2011.2132470","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.2011.2132470","pdf_url":null,"source":{"id":"https://openalex.org/S4502562","display_name":"IEEE Transactions on Information Theory","issn_l":"0018-9448","issn":["0018-9448","1557-9654"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Information Theory","raw_type":"journal-article"},{"id":"pmh:oai:mediatum.ub.tum.de:node/1081677","is_oa":true,"landing_page_url":"http://mediatum.ub.tum.de/node?id=1081677","pdf_url":null,"source":{"id":"https://openalex.org/S4306400453","display_name":"mediaTUM \u2013 the media and publications repository of the Technical University Munich (Technical University Munich)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I62916508","host_organization_name":"Technical University of Munich","host_organization_lineage":["https://openalex.org/I62916508"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"article"},{"id":"pmh:oai:publica.fraunhofer.de:publica/225169","is_oa":false,"landing_page_url":"https://publica.fraunhofer.de/handle/publica/225169","pdf_url":null,"source":{"id":"https://openalex.org/S4306400318","display_name":"Fraunhofer-Publica (Fraunhofer-Gesellschaft)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4923324","host_organization_name":"Fraunhofer-Gesellschaft","host_organization_lineage":["https://openalex.org/I4923324"],"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":"journal article"}],"best_oa_location":{"id":"pmh:oai:mediatum.ub.tum.de:node/1081677","is_oa":true,"landing_page_url":"http://mediatum.ub.tum.de/node?id=1081677","pdf_url":null,"source":{"id":"https://openalex.org/S4306400453","display_name":"mediaTUM \u2013 the media and publications repository of the Technical University Munich (Technical University Munich)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I62916508","host_organization_name":"Technical University of Munich","host_organization_lineage":["https://openalex.org/I62916508"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":53,"referenced_works":["https://openalex.org/W606886425","https://openalex.org/W1480550800","https://openalex.org/W1545861347","https://openalex.org/W1554220717","https://openalex.org/W1848804074","https://openalex.org/W1967045416","https://openalex.org/W1983097152","https://openalex.org/W1986592419","https://openalex.org/W2017990047","https://openalex.org/W2034716068","https://openalex.org/W2048419824","https://openalex.org/W2054244643","https://openalex.org/W2064509244","https://openalex.org/W2074722703","https://openalex.org/W2078896715","https://openalex.org/W2105726629","https://openalex.org/W2106809955","https://openalex.org/W2108251005","https://openalex.org/W2108321570","https://openalex.org/W2109121388","https://openalex.org/W2112820013","https://openalex.org/W2116407052","https://openalex.org/W2119854298","https://openalex.org/W2123740318","https://openalex.org/W2128296785","https://openalex.org/W2128978199","https://openalex.org/W2129772307","https://openalex.org/W2131431638","https://openalex.org/W2134595329","https://openalex.org/W2139518987","https://openalex.org/W2144778327","https://openalex.org/W2145977952","https://openalex.org/W2152701162","https://openalex.org/W2153937378","https://openalex.org/W2154779036","https://openalex.org/W2159489547","https://openalex.org/W2159715570","https://openalex.org/W2160243032","https://openalex.org/W2161297670","https://openalex.org/W2161473225","https://openalex.org/W2164271287","https://openalex.org/W2170131502","https://openalex.org/W2296319761","https://openalex.org/W2329742082","https://openalex.org/W2431771009","https://openalex.org/W3125878214","https://openalex.org/W4240896209","https://openalex.org/W4250589301","https://openalex.org/W4285719527","https://openalex.org/W4297970755","https://openalex.org/W6618594512","https://openalex.org/W6680828790","https://openalex.org/W6702258522"],"related_works":["https://openalex.org/W2356261519","https://openalex.org/W2026600343","https://openalex.org/W4382519084","https://openalex.org/W3092518081","https://openalex.org/W2391247224","https://openalex.org/W2002037426","https://openalex.org/W1978014154","https://openalex.org/W3020982396","https://openalex.org/W2353104181","https://openalex.org/W2040676043"],"abstract_inverted_index":{"Many":[0],"solutions":[1],"and":[2,7,44,82,114],"concepts":[3],"in":[4],"resource":[5],"allocation":[6],"game":[8],"theory":[9],"rely":[10],"on":[11,67,103],"the":[12,25,39,53,68,76,88,104,108,121],"assumption":[13,28,89],"of":[14,29,41,61,70,78,90,99,120],"a":[15,30,58,83],"convex":[16],"utility":[17],"set.":[18],"In":[19],"this":[20],"paper,":[21],"we":[22,94,111],"show":[23,95],"that":[24],"less":[26],"restrictive":[27],"logarithmic":[31],"\u201chidden\u201d":[32],"convexity":[33],"is":[34],"sometimes":[35],"sufficient.":[36],"We":[37,51,64],"consider":[38],"problems":[40],"Nash":[42,54],"bargaining":[43,55],"proportional":[45],"fairness,":[46],"which":[47],"are":[48],"closely":[49],"related.":[50],"extend":[52],"framework":[56],"to":[57],"broader":[59],"family":[60],"log-convex":[62,91],"sets.":[63],"then":[65],"focus":[66],"set":[69],"feasible":[71,122],"signal-to-interference-plus-noise":[72],"ratios":[73],"(SINRs),":[74],"for":[75,117],"cases":[77],"individual":[79],"power":[80,85],"constraints":[81],"sum":[84],"constraint.":[86],"Under":[87],"interference":[92,105],"functions,":[93],"how":[96],"Pareto":[97],"optimality":[98],"boundary":[100],"points":[101],"depends":[102],"coupling":[106],"between":[107],"users.":[109],"Finally,":[110],"provide":[112],"necessary":[113],"sufficient":[115],"conditions":[116],"strict":[118],"log-convexity":[119],"SINR":[123],"region.":[124]},"counts_by_year":[{"year":2022,"cited_by_count":2},{"year":2017,"cited_by_count":2},{"year":2016,"cited_by_count":2},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":6},{"year":2013,"cited_by_count":2}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}
