{"id":"https://openalex.org/W2165786861","doi":"https://doi.org/10.1109/tsp.2009.2016871","title":"Duality Gap Estimation and Polynomial Time Approximation for Optimal Spectrum Management","display_name":"Duality Gap Estimation and Polynomial Time Approximation for Optimal Spectrum Management","publication_year":2009,"publication_date":"2009-03-12","ids":{"openalex":"https://openalex.org/W2165786861","doi":"https://doi.org/10.1109/tsp.2009.2016871","mag":"2165786861"},"language":"en","primary_location":{"id":"doi:10.1109/tsp.2009.2016871","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tsp.2009.2016871","pdf_url":null,"source":{"id":"https://openalex.org/S168680287","display_name":"IEEE Transactions on Signal Processing","issn_l":"1053-587X","issn":["1053-587X","1941-0476"],"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 Signal Processing","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/A5101766226","display_name":"Zhi\u2010Quan Luo","orcid":"https://orcid.org/0000-0003-3995-914X"},"institutions":[{"id":"https://openalex.org/I130238516","display_name":"University of Minnesota","ror":"https://ror.org/017zqws13","country_code":"US","type":"education","lineage":["https://openalex.org/I130238516"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Zhi-Quan Luo","raw_affiliation_strings":["Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN, USA","institution_ids":["https://openalex.org/I130238516"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5089542160","display_name":"Shuzhong Zhang","orcid":"https://orcid.org/0000-0001-5838-1676"},"institutions":[{"id":"https://openalex.org/I177725633","display_name":"Chinese University of Hong Kong","ror":"https://ror.org/00t33hh48","country_code":"HK","type":"education","lineage":["https://openalex.org/I177725633"]}],"countries":["HK"],"is_corresponding":false,"raw_author_name":"Shuzhong Zhang","raw_affiliation_strings":["Department of Systems Engineering and Engineering Management, Chinese University of Hong Kong, Sha Tin, Hong Kong, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Systems Engineering and Engineering Management, Chinese University of Hong Kong, Sha Tin, Hong Kong, China","institution_ids":["https://openalex.org/I177725633"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":5.0234,"has_fulltext":false,"cited_by_count":51,"citation_normalized_percentile":{"value":0.95375741,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"57","issue":"7","first_page":"2675","last_page":"2689"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10148","display_name":"Advanced MIMO Systems Optimization","score":0.9998000264167786,"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":0.9998000264167786,"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/T10579","display_name":"Cognitive Radio Networks and Spectrum Sensing","score":0.9993000030517578,"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"}},{"id":"https://openalex.org/T11409","display_name":"Advanced Wireless Network Optimization","score":0.9983000159263611,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/duality","display_name":"Duality (order theory)","score":0.6205964684486389},{"id":"https://openalex.org/keywords/duality-gap","display_name":"Duality gap","score":0.5772698521614075},{"id":"https://openalex.org/keywords/discretization","display_name":"Discretization","score":0.5532153844833374},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.4256259799003601},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.3858039379119873},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.3635860085487366},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.3238255977630615},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3174762725830078},{"id":"https://openalex.org/keywords/optimization-problem","display_name":"Optimization problem","score":0.3164668679237366},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.2040480673313141}],"concepts":[{"id":"https://openalex.org/C2778023678","wikidata":"https://www.wikidata.org/wiki/Q554403","display_name":"Duality (order theory)","level":2,"score":0.6205964684486389},{"id":"https://openalex.org/C5274546","wikidata":"https://www.wikidata.org/wiki/Q5310264","display_name":"Duality gap","level":3,"score":0.5772698521614075},{"id":"https://openalex.org/C73000952","wikidata":"https://www.wikidata.org/wiki/Q17007827","display_name":"Discretization","level":2,"score":0.5532153844833374},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.4256259799003601},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3858039379119873},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3635860085487366},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.3238255977630615},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3174762725830078},{"id":"https://openalex.org/C137836250","wikidata":"https://www.wikidata.org/wiki/Q984063","display_name":"Optimization problem","level":2,"score":0.3164668679237366},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.2040480673313141}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1109/tsp.2009.2016871","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tsp.2009.2016871","pdf_url":null,"source":{"id":"https://openalex.org/S168680287","display_name":"IEEE Transactions on Signal Processing","issn_l":"1053-587X","issn":["1053-587X","1941-0476"],"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 Signal Processing","raw_type":"journal-article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.555.6916","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.555.6916","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.ece.umn.edu/~luozq/assets/pdf/publications_files/Luo_DualityGap.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.700.9002","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.700.9002","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.menet.umn.edu/%7Ezhangs/Reports/2009_LZ.pdf","raw_type":"text"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":31,"referenced_works":["https://openalex.org/W294645195","https://openalex.org/W1585828867","https://openalex.org/W1967073510","https://openalex.org/W1989873462","https://openalex.org/W2000354435","https://openalex.org/W2002937495","https://openalex.org/W2019657104","https://openalex.org/W2047376401","https://openalex.org/W2056107761","https://openalex.org/W2059179067","https://openalex.org/W2071707134","https://openalex.org/W2077242588","https://openalex.org/W2091237335","https://openalex.org/W2099111195","https://openalex.org/W2110391887","https://openalex.org/W2124409123","https://openalex.org/W2130156169","https://openalex.org/W2139219777","https://openalex.org/W2144000019","https://openalex.org/W2144145247","https://openalex.org/W2145203050","https://openalex.org/W2153344215","https://openalex.org/W2154512502","https://openalex.org/W2161272050","https://openalex.org/W2280409710","https://openalex.org/W2479293754","https://openalex.org/W4235381922","https://openalex.org/W4248623816","https://openalex.org/W4249667877","https://openalex.org/W6647672507","https://openalex.org/W6681297494"],"related_works":["https://openalex.org/W2593555778","https://openalex.org/W2201555914","https://openalex.org/W2725387418","https://openalex.org/W2614523486","https://openalex.org/W640885572","https://openalex.org/W4220928172","https://openalex.org/W2089739134","https://openalex.org/W2046743339","https://openalex.org/W228544604","https://openalex.org/W2000354435"],"abstract_inverted_index":{"Consider":[0],"a":[1,8,39,94,103,174],"communication":[2],"system":[3],"whereby":[4],"multiple":[5],"users":[6,35],"share":[7],"common":[9],"frequency":[10,147],"band":[11],"and":[12,79,121],"must":[13],"choose":[14],"their":[15],"transmit":[16],"power":[17,52],"spectra":[18],"jointly":[19],"in":[20,84,191],"response":[21],"to":[22,37,50,57,88,93,108,159],"physical":[23],"channel":[24],"conditions":[25],"including":[26],"the":[27,34,59,72,89,110,116,124,130,133,136,140,144,149,160,168,183,187],"effects":[28],"of":[29,33,46,62,106,112,135,139,186],"interference.":[30],"The":[31],"goal":[32],"is":[36,66,77,83],"maximize":[38],"system-wide":[40],"utility":[41],"function":[42],"(e.g.,":[43],"weighted":[44],"sum-rate":[45],"all":[47],"users),":[48],"subject":[49],"individual":[51],"constraints.":[53],"A":[54],"popular":[55],"approach":[56],"solve":[58],"discretized":[60,73,117],"version":[61],"this":[63,99],"nonconvex":[64],"problem":[65,76,120,190],"by":[67],"Lagrangian":[68,81,169],"dual":[69,82,170],"relaxation.":[70],"Unfortunately":[71],"spectrum":[74,118,162,188],"management":[75,119,189],"NP-hard":[78],"its":[80],"general":[85],"not":[86],"equivalent":[87],"primal":[90],"formulation":[91,185],"due":[92],"positive":[95],"duality":[96,113,125,150],"gap.":[97],"In":[98],"paper,":[100],"we":[101,165],"use":[102],"convexity":[104],"result":[105],"Lyapunov":[107],"estimate":[109,152],"size":[111,134],"gap":[114,126,151],"for":[115,182,194],"show":[122,166],"that":[123,167],"vanishes":[127],"asymptotically":[128],"at":[129],"rateO(1/radicN),":[131],"whereNis":[132],"uniform":[137],"discretization":[138],"shared":[141],"spectrum.":[142],"If":[143],"channels":[145],"are":[146],"flat,":[148],"improves":[153],"toO(1/N)":[154],".":[155],"Moreover,":[156],"when":[157],"restricted":[158],"FDMA":[161],"sharing":[163],"strategies,":[164],"relaxation,":[171],"combined":[172],"with":[173],"linear":[175],"programming":[176],"scheme,":[177],"can":[178],"generate":[179],"anepsiv-optimal":[180],"solution":[181],"continuous":[184],"polynomial":[192],"time":[193],"anyepsiv>0.":[195]},"counts_by_year":[{"year":2023,"cited_by_count":1},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":3},{"year":2018,"cited_by_count":2},{"year":2017,"cited_by_count":1},{"year":2016,"cited_by_count":4},{"year":2015,"cited_by_count":4},{"year":2014,"cited_by_count":10},{"year":2013,"cited_by_count":5},{"year":2012,"cited_by_count":6}],"updated_date":"2026-07-29T09:40:50.615796","created_date":"2025-10-10T00:00:00"}