{"id":"https://openalex.org/W2059607526","doi":"https://doi.org/10.1109/tsp.2013.2245126","title":"Duality and Optimization for Generalized Multi-Hop MIMO Amplify-and-Forward Relay Networks With Linear Constraints","display_name":"Duality and Optimization for Generalized Multi-Hop MIMO Amplify-and-Forward Relay Networks With Linear Constraints","publication_year":2013,"publication_date":"2013-02-05","ids":{"openalex":"https://openalex.org/W2059607526","doi":"https://doi.org/10.1109/tsp.2013.2245126","mag":"2059607526"},"language":"en","primary_location":{"id":"doi:10.1109/tsp.2013.2245126","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tsp.2013.2245126","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":["arxiv","crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1301.4729","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5100340369","display_name":"An Liu","orcid":"https://orcid.org/0000-0002-3943-5234"},"institutions":[{"id":"https://openalex.org/I200769079","display_name":"Hong Kong University of Science and Technology","ror":"https://ror.org/00q4vv597","country_code":"HK","type":"education","lineage":["https://openalex.org/I200769079"]}],"countries":["HK"],"is_corresponding":false,"raw_author_name":"An Liu","raw_affiliation_strings":["Department of Electrical and Computer Engineering, Hong Kong University of Science and Technology, Hong Kong, China","Dept. of Electr. & Comput. Eng., Hong Kong Univ. of Sci. & Technol., Hong Kong, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Electrical and Computer Engineering, Hong Kong University of Science and Technology, Hong Kong, China","institution_ids":["https://openalex.org/I200769079"]},{"raw_affiliation_string":"Dept. of Electr. & Comput. Eng., Hong Kong Univ. of Sci. & Technol., Hong Kong, China","institution_ids":["https://openalex.org/I200769079"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5073153992","display_name":"Vincent K. N. Lau","orcid":"https://orcid.org/0000-0001-7769-6008"},"institutions":[{"id":"https://openalex.org/I200769079","display_name":"Hong Kong University of Science and Technology","ror":"https://ror.org/00q4vv597","country_code":"HK","type":"education","lineage":["https://openalex.org/I200769079"]}],"countries":["HK"],"is_corresponding":false,"raw_author_name":"Vincent K. N. Lau","raw_affiliation_strings":["Department of Electrical and Computer Engineering, Hong Kong University of Science and Technology, Hong Kong, China","Dept. of Electr. & Comput. Eng., Hong Kong Univ. of Sci. & Technol., Hong Kong, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Electrical and Computer Engineering, Hong Kong University of Science and Technology, Hong Kong, China","institution_ids":["https://openalex.org/I200769079"]},{"raw_affiliation_string":"Dept. of Electr. & Comput. Eng., Hong Kong Univ. of Sci. & Technol., Hong Kong, China","institution_ids":["https://openalex.org/I200769079"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5084454849","display_name":"Youjian Liu","orcid":"https://orcid.org/0000-0003-4681-0546"},"institutions":[{"id":"https://openalex.org/I188538660","display_name":"University of Colorado Boulder","ror":"https://ror.org/02ttsq026","country_code":"US","type":"education","lineage":["https://openalex.org/I188538660"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Youjian Liu","raw_affiliation_strings":["Department of Electrical, Computer, and Energy Engineering, University of Colorado, Boulder, Boulder, CO, USA","Dept. of Electr. Comput. & Energy Eng., Univ. of Colorado at Boulder, Boulder, CO, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Electrical, Computer, and Energy Engineering, University of Colorado, Boulder, Boulder, CO, USA","institution_ids":["https://openalex.org/I188538660"]},{"raw_affiliation_string":"Dept. of Electr. Comput. & Energy Eng., Univ. of Colorado at Boulder, Boulder, CO, USA","institution_ids":["https://openalex.org/I188538660"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":3.2942,"has_fulltext":false,"cited_by_count":16,"citation_normalized_percentile":{"value":0.92766646,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"61","issue":"9","first_page":"2356","last_page":"2365"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10796","display_name":"Cooperative Communication and Network Coding","score":1.0,"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"}},"topics":[{"id":"https://openalex.org/T10796","display_name":"Cooperative Communication and Network Coding","score":1.0,"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/T12791","display_name":"Full-Duplex Wireless Communications","score":0.9980999827384949,"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/T10148","display_name":"Advanced MIMO Systems Optimization","score":0.9962000250816345,"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/mathematical-optimization","display_name":"Mathematical optimization","score":0.6679525375366211},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5773199200630188},{"id":"https://openalex.org/keywords/optimization-problem","display_name":"Optimization problem","score":0.5261784791946411},{"id":"https://openalex.org/keywords/duality","display_name":"Duality (order theory)","score":0.5201883316040039},{"id":"https://openalex.org/keywords/relay","display_name":"Relay","score":0.5095870494842529},{"id":"https://openalex.org/keywords/mimo","display_name":"MIMO","score":0.48984023928642273},{"id":"https://openalex.org/keywords/stationary-point","display_name":"Stationary point","score":0.4626081883907318},{"id":"https://openalex.org/keywords/strong-duality","display_name":"Strong duality","score":0.45960357785224915},{"id":"https://openalex.org/keywords/inner-loop","display_name":"Inner loop","score":0.45350930094718933},{"id":"https://openalex.org/keywords/duality-gap","display_name":"Duality gap","score":0.44114112854003906},{"id":"https://openalex.org/keywords/linear-programming","display_name":"Linear programming","score":0.43490681052207947},{"id":"https://openalex.org/keywords/interior-point-method","display_name":"Interior point method","score":0.4131471514701843},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.11427038908004761}],"concepts":[{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.6679525375366211},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5773199200630188},{"id":"https://openalex.org/C137836250","wikidata":"https://www.wikidata.org/wiki/Q984063","display_name":"Optimization problem","level":2,"score":0.5261784791946411},{"id":"https://openalex.org/C2778023678","wikidata":"https://www.wikidata.org/wiki/Q554403","display_name":"Duality (order theory)","level":2,"score":0.5201883316040039},{"id":"https://openalex.org/C2778156585","wikidata":"https://www.wikidata.org/wiki/Q174053","display_name":"Relay","level":3,"score":0.5095870494842529},{"id":"https://openalex.org/C207987634","wikidata":"https://www.wikidata.org/wiki/Q176862","display_name":"MIMO","level":3,"score":0.48984023928642273},{"id":"https://openalex.org/C189237950","wikidata":"https://www.wikidata.org/wiki/Q2500758","display_name":"Stationary point","level":2,"score":0.4626081883907318},{"id":"https://openalex.org/C163863214","wikidata":"https://www.wikidata.org/wiki/Q7624553","display_name":"Strong duality","level":3,"score":0.45960357785224915},{"id":"https://openalex.org/C58716799","wikidata":"https://www.wikidata.org/wiki/Q6035648","display_name":"Inner loop","level":3,"score":0.45350930094718933},{"id":"https://openalex.org/C5274546","wikidata":"https://www.wikidata.org/wiki/Q5310264","display_name":"Duality gap","level":3,"score":0.44114112854003906},{"id":"https://openalex.org/C41045048","wikidata":"https://www.wikidata.org/wiki/Q202843","display_name":"Linear programming","level":2,"score":0.43490681052207947},{"id":"https://openalex.org/C155253501","wikidata":"https://www.wikidata.org/wiki/Q461992","display_name":"Interior point method","level":2,"score":0.4131471514701843},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.11427038908004761},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C54197355","wikidata":"https://www.wikidata.org/wiki/Q5782992","display_name":"Beamforming","level":2,"score":0.0},{"id":"https://openalex.org/C203479927","wikidata":"https://www.wikidata.org/wiki/Q5165939","display_name":"Controller (irrigation)","level":2,"score":0.0},{"id":"https://openalex.org/C6557445","wikidata":"https://www.wikidata.org/wiki/Q173113","display_name":"Agronomy","level":1,"score":0.0},{"id":"https://openalex.org/C163258240","wikidata":"https://www.wikidata.org/wiki/Q25342","display_name":"Power (physics)","level":2,"score":0.0},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","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},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1109/tsp.2013.2245126","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tsp.2013.2245126","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:arXiv.org:1301.4729","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1301.4729","pdf_url":"https://arxiv.org/pdf/1301.4729","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"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":"text"},{"id":"pmh:oai:repository.hkust.edu.hk:1783.1-57452","is_oa":false,"landing_page_url":"http://repository.hkust.edu.hk/ir/Record/1783.1-57452","pdf_url":null,"source":{"id":"https://openalex.org/S4306401796","display_name":"Rare & Special e-Zone (The Hong Kong 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/I200769079","host_organization_name":"Hong Kong University of Science and Technology","host_organization_lineage":["https://openalex.org/I200769079"],"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":"pmh:oai:arXiv.org:1301.4729","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1301.4729","pdf_url":"https://arxiv.org/pdf/1301.4729","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"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":"text"},"sustainable_development_goals":[{"score":0.75,"id":"https://metadata.un.org/sdg/6","display_name":"Clean water and sanitation"}],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":29,"referenced_works":["https://openalex.org/W1487444668","https://openalex.org/W1822442307","https://openalex.org/W1976109068","https://openalex.org/W1982472349","https://openalex.org/W2034771294","https://openalex.org/W2037869274","https://openalex.org/W2059607526","https://openalex.org/W2067145564","https://openalex.org/W2075870768","https://openalex.org/W2076853942","https://openalex.org/W2113026489","https://openalex.org/W2121342401","https://openalex.org/W2124412092","https://openalex.org/W2124887991","https://openalex.org/W2130509920","https://openalex.org/W2130613910","https://openalex.org/W2142383973","https://openalex.org/W2148006752","https://openalex.org/W2153356113","https://openalex.org/W2154119053","https://openalex.org/W2164171933","https://openalex.org/W2167249094","https://openalex.org/W2168784354","https://openalex.org/W2296319761","https://openalex.org/W4242209908","https://openalex.org/W4250589301","https://openalex.org/W4291236916","https://openalex.org/W4301621763","https://openalex.org/W6638772164"],"related_works":["https://openalex.org/W2155425983","https://openalex.org/W1533868172","https://openalex.org/W2089042254","https://openalex.org/W4220928172","https://openalex.org/W2725387418","https://openalex.org/W2025430946","https://openalex.org/W2056178476","https://openalex.org/W2614523486","https://openalex.org/W2763044528","https://openalex.org/W1997572981"],"abstract_inverted_index":{"We":[0,18],"consider":[1],"a":[2,29,51,68,79,103,108],"generalized":[3],"multi-hop":[4],"MIMO":[5],"amplify-and-forward":[6],"(AF)":[7],"relay":[8,64],"network":[9,30],"with":[10,129,137,149],"multiple":[11],"sources/destinations":[12],"and":[13,22,31,39,118],"arbitrarily":[14],"number":[15],"of":[16,58,62,87,111],"relays.":[17],"establish":[19],"two":[20],"dualities":[21],"the":[23,74,83,89,92,112,130,133],"corresponding":[24],"dual":[25,71],"transformations":[26],"between":[27],"such":[28],"its":[32],"dual,":[33],"respectively,":[34],"under":[35],"single-network":[36],"linear":[37,41],"constraint":[38],"per-hop":[40],"constraint.":[42],"A":[43],"unified":[44],"optimization":[45,60,135],"framework":[46,136],"is":[47],"proposed":[48,134],"to":[49,101,123,144],"find":[50],"stationary":[52,80,109],"point":[53,81,110],"for":[54,82],"an":[55],"important":[56],"class":[57],"nonconvex":[59],"problems":[61,148],"AF":[63],"networks":[65],"based":[66],"on":[67],"local":[69],"Lagrange":[70],"method,":[72],"where":[73],"primal":[75,93,126,139],"algorithm":[76],"only":[77],"finds":[78],"inner":[84,113],"loop":[85,114],"problem":[86],"maximizing":[88],"Lagrangian":[90],"w.r.t.":[91],"variables.":[94],"The":[95,116],"input":[96],"covariance":[97],"matrices":[98],"are":[99,121],"shown":[100],"satisfy":[102],"polite":[104,119],"water-filling":[105,120],"structure":[106],"at":[107],"problem.":[115],"duality":[117],"exploited":[122],"design":[124],"fast":[125],"algorithms.":[127],"Compared":[128],"existing":[131],"algorithms,":[132],"duality-based":[138],"algorithms":[140],"can":[141],"be":[142],"used":[143],"solve":[145],"more":[146],"general":[147],"lower":[150],"computation":[151],"cost.":[152]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":2},{"year":2019,"cited_by_count":1},{"year":2017,"cited_by_count":2},{"year":2016,"cited_by_count":3},{"year":2014,"cited_by_count":4},{"year":2013,"cited_by_count":2}],"updated_date":"2026-07-03T08:13:44.112507","created_date":"2016-06-24T00:00:00"}