{"id":"https://openalex.org/W2117068724","doi":"https://doi.org/10.1162/evco_a_00157","title":"Hypervolume Subset Selection in Two Dimensions: Formulations and Algorithms","display_name":"Hypervolume Subset Selection in Two Dimensions: Formulations and Algorithms","publication_year":2015,"publication_date":"2015-07-02","ids":{"openalex":"https://openalex.org/W2117068724","doi":"https://doi.org/10.1162/evco_a_00157","mag":"2117068724","pmid":"https://pubmed.ncbi.nlm.nih.gov/26135717"},"language":"en","primary_location":{"id":"doi:10.1162/evco_a_00157","is_oa":false,"landing_page_url":"https://doi.org/10.1162/evco_a_00157","pdf_url":null,"source":{"id":"https://openalex.org/S38677346","display_name":"Evolutionary Computation","issn_l":"1063-6560","issn":["1063-6560","1530-9304"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315718","host_organization_name":"The MIT Press","host_organization_lineage":["https://openalex.org/P4310315718"],"host_organization_lineage_names":["The MIT Press"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Evolutionary Computation","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","pubmed"],"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/A5041404272","display_name":"Tobias K\u00fchn","orcid":"https://orcid.org/0000-0003-0741-1083"},"institutions":[{"id":"https://openalex.org/I153267046","display_name":"University of Kaiserslautern","ror":"https://ror.org/04zrf7b53","country_code":"DE","type":"education","lineage":["https://openalex.org/I153267046"]}],"countries":["DE"],"is_corresponding":true,"raw_author_name":"Tobias Kuhn","raw_affiliation_strings":["Mathematical Institute, University of Kaiserslautern, Germany"],"affiliations":[{"raw_affiliation_string":"Mathematical Institute, University of Kaiserslautern, Germany","institution_ids":["https://openalex.org/I153267046"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5038504776","display_name":"Carlos M. Fonseca","orcid":"https://orcid.org/0000-0001-5162-2457"},"institutions":[{"id":"https://openalex.org/I76903346","display_name":"University of Coimbra","ror":"https://ror.org/04z8k9a98","country_code":"PT","type":"education","lineage":["https://openalex.org/I76903346"]}],"countries":["PT"],"is_corresponding":false,"raw_author_name":"Carlos M. Fonseca","raw_affiliation_strings":["CISUC, Department of Informatics Engineering, University of Coimbra, Portugal","CISUC, Department of Informatics Engineering, University of Coimbra (Portugal)"],"affiliations":[{"raw_affiliation_string":"CISUC, Department of Informatics Engineering, University of Coimbra, Portugal","institution_ids":["https://openalex.org/I76903346"]},{"raw_affiliation_string":"CISUC, Department of Informatics Engineering, University of Coimbra (Portugal)","institution_ids":["https://openalex.org/I76903346"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5055996291","display_name":"Lu\u00eds Paquete","orcid":"https://orcid.org/0000-0001-7525-8901"},"institutions":[{"id":"https://openalex.org/I76903346","display_name":"University of Coimbra","ror":"https://ror.org/04z8k9a98","country_code":"PT","type":"education","lineage":["https://openalex.org/I76903346"]}],"countries":["PT"],"is_corresponding":false,"raw_author_name":"Lu\u00eds Paquete","raw_affiliation_strings":["CISUC, Department of Informatics Engineering, University of Coimbra, Portugal","CISUC, Department of Informatics Engineering, University of Coimbra (Portugal)"],"affiliations":[{"raw_affiliation_string":"CISUC, Department of Informatics Engineering, University of Coimbra, Portugal","institution_ids":["https://openalex.org/I76903346"]},{"raw_affiliation_string":"CISUC, Department of Informatics Engineering, University of Coimbra (Portugal)","institution_ids":["https://openalex.org/I76903346"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5107846021","display_name":"Stefan Ruzika","orcid":null},"institutions":[{"id":"https://openalex.org/I4387153937","display_name":"Universit\u00e4t Koblenz","ror":"https://ror.org/0433e6t24","country_code":null,"type":"education","lineage":["https://openalex.org/I4387153937"]},{"id":"https://openalex.org/I2802076133","display_name":"University of Koblenz and Landau","ror":"https://ror.org/01j9f6752","country_code":"DE","type":"education","lineage":["https://openalex.org/I2802076133"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Stefan Ruzika","raw_affiliation_strings":["Mathematical Institute, University of Koblenz-Landau, Campus Koblenz, Germany"],"affiliations":[{"raw_affiliation_string":"Mathematical Institute, University of Koblenz-Landau, Campus Koblenz, Germany","institution_ids":["https://openalex.org/I2802076133","https://openalex.org/I4387153937"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5067685798","display_name":"Miguel M. Duarte","orcid":null},"institutions":[{"id":"https://openalex.org/I76903346","display_name":"University of Coimbra","ror":"https://ror.org/04z8k9a98","country_code":"PT","type":"education","lineage":["https://openalex.org/I76903346"]}],"countries":["PT"],"is_corresponding":false,"raw_author_name":"Miguel M. Duarte","raw_affiliation_strings":["CISUC, Department of Informatics Engineering, University of Coimbra, Portugal","CISUC, Department of Informatics Engineering, University of Coimbra (Portugal)"],"affiliations":[{"raw_affiliation_string":"CISUC, Department of Informatics Engineering, University of Coimbra, Portugal","institution_ids":["https://openalex.org/I76903346"]},{"raw_affiliation_string":"CISUC, Department of Informatics Engineering, University of Coimbra (Portugal)","institution_ids":["https://openalex.org/I76903346"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5024152051","display_name":"Jos\u00e9 Rui Figueira","orcid":"https://orcid.org/0000-0001-9822-1167"},"institutions":[{"id":"https://openalex.org/I203847022","display_name":"Instituto Polit\u00e9cnico de Lisboa","ror":"https://ror.org/04ea70f07","country_code":"PT","type":"education","lineage":["https://openalex.org/I203847022"]},{"id":"https://openalex.org/I141596103","display_name":"University of Lisbon","ror":"https://ror.org/01c27hj86","country_code":"PT","type":"education","lineage":["https://openalex.org/I141596103"]}],"countries":["PT"],"is_corresponding":false,"raw_author_name":"Jos\u00e9 Rui Figueira","raw_affiliation_strings":["CEG-IST, Instituto Superior T\u00e9cnico, Universidade de Lisboa, Portugal"],"affiliations":[{"raw_affiliation_string":"CEG-IST, Instituto Superior T\u00e9cnico, Universidade de Lisboa, Portugal","institution_ids":["https://openalex.org/I203847022","https://openalex.org/I141596103"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":6,"corresponding_author_ids":["https://openalex.org/A5041404272"],"corresponding_institution_ids":["https://openalex.org/I153267046"],"apc_list":null,"apc_paid":null,"fwci":7.6292,"has_fulltext":false,"cited_by_count":73,"citation_normalized_percentile":{"value":0.97795063,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":94,"max":99},"biblio":{"volume":"24","issue":"3","first_page":"411","last_page":"425"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10848","display_name":"Advanced Multi-Objective Optimization Algorithms","score":0.9997000098228455,"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/T10848","display_name":"Advanced Multi-Objective Optimization Algorithms","score":0.9997000098228455,"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/T10791","display_name":"Advanced Control Systems Optimization","score":0.9983999729156494,"subfield":{"id":"https://openalex.org/subfields/2207","display_name":"Control and Systems 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/T11053","display_name":"Process Optimization and Integration","score":0.9879000186920166,"subfield":{"id":"https://openalex.org/subfields/2207","display_name":"Control and Systems 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/mathematics","display_name":"Mathematics","score":0.7007037997245789},{"id":"https://openalex.org/keywords/selection","display_name":"Selection (genetic algorithm)","score":0.6653754711151123},{"id":"https://openalex.org/keywords/digraph","display_name":"Digraph","score":0.6184309720993042},{"id":"https://openalex.org/keywords/cardinality","display_name":"Cardinality (data modeling)","score":0.6097615361213684},{"id":"https://openalex.org/keywords/linear-programming","display_name":"Linear programming","score":0.5520642995834351},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.5483936071395874},{"id":"https://openalex.org/keywords/relaxation","display_name":"Relaxation (psychology)","score":0.5272782444953918},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.5053701996803284},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.45072224736213684},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.448578417301178},{"id":"https://openalex.org/keywords/integer-programming","display_name":"Integer programming","score":0.43216004967689514},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.2495517134666443}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7007037997245789},{"id":"https://openalex.org/C81917197","wikidata":"https://www.wikidata.org/wiki/Q628760","display_name":"Selection (genetic algorithm)","level":2,"score":0.6653754711151123},{"id":"https://openalex.org/C2779145032","wikidata":"https://www.wikidata.org/wiki/Q191494","display_name":"Digraph","level":2,"score":0.6184309720993042},{"id":"https://openalex.org/C87117476","wikidata":"https://www.wikidata.org/wiki/Q362383","display_name":"Cardinality (data modeling)","level":2,"score":0.6097615361213684},{"id":"https://openalex.org/C41045048","wikidata":"https://www.wikidata.org/wiki/Q202843","display_name":"Linear programming","level":2,"score":0.5520642995834351},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.5483936071395874},{"id":"https://openalex.org/C2776029896","wikidata":"https://www.wikidata.org/wiki/Q3935810","display_name":"Relaxation (psychology)","level":2,"score":0.5272782444953918},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.5053701996803284},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.45072224736213684},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.448578417301178},{"id":"https://openalex.org/C56086750","wikidata":"https://www.wikidata.org/wiki/Q6042592","display_name":"Integer programming","level":2,"score":0.43216004967689514},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.2495517134666443},{"id":"https://openalex.org/C77805123","wikidata":"https://www.wikidata.org/wiki/Q161272","display_name":"Social psychology","level":1,"score":0.0},{"id":"https://openalex.org/C124101348","wikidata":"https://www.wikidata.org/wiki/Q172491","display_name":"Data mining","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/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.0},{"id":"https://openalex.org/C15744967","wikidata":"https://www.wikidata.org/wiki/Q9418","display_name":"Psychology","level":0,"score":0.0}],"mesh":[{"descriptor_ui":"D000465","descriptor_name":"Algorithms","qualifier_ui":null,"qualifier_name":null,"is_major_topic":true},{"descriptor_ui":"D000465","descriptor_name":"Algorithms","qualifier_ui":null,"qualifier_name":null,"is_major_topic":true},{"descriptor_ui":"D000465","descriptor_name":"Algorithms","qualifier_ui":null,"qualifier_name":null,"is_major_topic":true},{"descriptor_ui":"D008962","descriptor_name":"Models, Theoretical","qualifier_ui":null,"qualifier_name":null,"is_major_topic":true},{"descriptor_ui":"D008962","descriptor_name":"Models, Theoretical","qualifier_ui":null,"qualifier_name":null,"is_major_topic":true},{"descriptor_ui":"D008962","descriptor_name":"Models, Theoretical","qualifier_ui":null,"qualifier_name":null,"is_major_topic":true}],"locations_count":6,"locations":[{"id":"doi:10.1162/evco_a_00157","is_oa":false,"landing_page_url":"https://doi.org/10.1162/evco_a_00157","pdf_url":null,"source":{"id":"https://openalex.org/S38677346","display_name":"Evolutionary Computation","issn_l":"1063-6560","issn":["1063-6560","1530-9304"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315718","host_organization_name":"The MIT Press","host_organization_lineage":["https://openalex.org/P4310315718"],"host_organization_lineage_names":["The MIT Press"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Evolutionary Computation","raw_type":"journal-article"},{"id":"pmid:26135717","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/26135717","pdf_url":null,"source":{"id":"https://openalex.org/S4306525036","display_name":"PubMed","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Evolutionary computation","raw_type":null},{"id":"pmh:oai:kluedo.ub.rptu.de:3770","is_oa":false,"landing_page_url":"https://nbn-resolving.org/urn:nbn:de:hbz:386-kluedo-37700","pdf_url":null,"source":{"id":"https://openalex.org/S4306402046","display_name":"Publication Server of Kaiserslautern University of Technology (Kaiserslautern University of Technology)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I153267046","host_organization_name":"University of Kaiserslautern","host_organization_lineage":["https://openalex.org/I153267046"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"preprint"},{"id":"pmh:oai:kluedo.ub.rptu.de:3798","is_oa":false,"landing_page_url":"https://nbn-resolving.org/urn:nbn:de:hbz:386-kluedo-37983","pdf_url":null,"source":{"id":"https://openalex.org/S4306402046","display_name":"Publication Server of Kaiserslautern University of Technology (Kaiserslautern University of Technology)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I153267046","host_organization_name":"University of Kaiserslautern","host_organization_lineage":["https://openalex.org/I153267046"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"preprint"},{"id":"pmh:oai:kluedo.ub.uni-kl.de:3770","is_oa":false,"landing_page_url":"https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3770","pdf_url":null,"source":{"id":"https://openalex.org/S4306402046","display_name":"Publication Server of Kaiserslautern University of Technology (Kaiserslautern University of Technology)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I153267046","host_organization_name":"University of Kaiserslautern","host_organization_lineage":["https://openalex.org/I153267046"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"preprint"},{"id":"pmh:oai:kluedo.ub.uni-kl.de:3798","is_oa":false,"landing_page_url":"https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3798","pdf_url":null,"source":{"id":"https://openalex.org/S4306402046","display_name":"Publication Server of Kaiserslautern University of Technology (Kaiserslautern University of Technology)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I153267046","host_organization_name":"University of Kaiserslautern","host_organization_lineage":["https://openalex.org/I153267046"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"preprint"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":38,"referenced_works":["https://openalex.org/W86912879","https://openalex.org/W118285713","https://openalex.org/W133511943","https://openalex.org/W260397477","https://openalex.org/W1480517346","https://openalex.org/W1491287429","https://openalex.org/W1521316934","https://openalex.org/W1549512796","https://openalex.org/W1558919105","https://openalex.org/W1588375755","https://openalex.org/W1613794742","https://openalex.org/W1968143987","https://openalex.org/W2008049315","https://openalex.org/W2010997912","https://openalex.org/W2016984768","https://openalex.org/W2024353273","https://openalex.org/W2029496228","https://openalex.org/W2038420231","https://openalex.org/W2039191483","https://openalex.org/W2055257826","https://openalex.org/W2064823886","https://openalex.org/W2098907614","https://openalex.org/W2108968575","https://openalex.org/W2118028531","https://openalex.org/W2123485079","https://openalex.org/W2125899728","https://openalex.org/W2129392074","https://openalex.org/W2136370773","https://openalex.org/W2154781286","https://openalex.org/W2160088187","https://openalex.org/W2165626989","https://openalex.org/W2165968314","https://openalex.org/W2172122763","https://openalex.org/W2463564488","https://openalex.org/W2504494713","https://openalex.org/W2798566347","https://openalex.org/W4233783227","https://openalex.org/W4237954928"],"related_works":["https://openalex.org/W2038088825","https://openalex.org/W1975203667","https://openalex.org/W2370328482","https://openalex.org/W4287882779","https://openalex.org/W4285278887","https://openalex.org/W2995925505","https://openalex.org/W2365237642","https://openalex.org/W3116484972","https://openalex.org/W2941986668","https://openalex.org/W4232431455"],"abstract_inverted_index":{"The":[0,60,78],"hypervolume":[1,24],"subset":[2],"selection":[3,30],"problem":[4,27],"consists":[5],"of":[6,15,18,32,57,112,127,155],"finding":[7],"a":[8,11,16,63,82,88],"subset,":[9],"with":[10,91],"given":[12],"cardinality":[13],"k,":[14],"set":[17],"nondominated":[19],"points":[20],"that":[21,68,95],"maximizes":[22],"the":[23,54,92,110,125],"indicator.":[25],"This":[26,107],"arises":[28],"in":[29,102,116,131],"procedures":[31],"evolutionary":[33],"algorithms":[34,42],"for":[35,38,53,152],"multiobjective":[36],"optimization,":[37],"which":[39,138],"practically":[40],"efficient":[41],"are":[43,51,150],"required.":[44],"In":[45],"this":[46,58,143],"article,":[47],"two":[48],"new":[49],"formulations":[50],"provided":[52],"two-dimensional":[55],"variant":[56],"problem.":[59],"first":[61],"is":[62,81],"(linear)":[64],"integer":[65],"programming":[66,76,101],"formulation":[67,80,86],"can":[69,96],"be":[70,97],"solved":[71,98],"by":[72,99],"solving":[73],"its":[74],"linear":[75],"relaxation.":[77],"second":[79],"k-link":[83],"shortest":[84],"path":[85],"on":[87],"special":[89],"digraph":[90],"Monge":[93],"property":[94],"dynamic":[100],"[Formula:":[103,113,128],"see":[104,114,129],"text]":[105,115,130],"time.":[106],"improves":[108,123],"upon":[109,124],"result":[111,126],"Bader":[117],"(":[118,135],"2009":[119],"),":[120,137],"and":[121,157],"slightly":[122],"Bringmann":[132],"et":[133],"al.":[134],"2014b":[136],"was":[139],"developed":[140],"independently":[141],"from":[142],"work":[144],"using":[145],"different":[146],"techniques.":[147],"Numerical":[148],"results":[149],"shown":[151],"several":[153],"values":[154],"n":[156],"k.":[158]},"counts_by_year":[{"year":2025,"cited_by_count":4},{"year":2024,"cited_by_count":5},{"year":2023,"cited_by_count":4},{"year":2022,"cited_by_count":4},{"year":2021,"cited_by_count":12},{"year":2020,"cited_by_count":14},{"year":2019,"cited_by_count":7},{"year":2018,"cited_by_count":6},{"year":2017,"cited_by_count":6},{"year":2016,"cited_by_count":4},{"year":2015,"cited_by_count":5},{"year":2014,"cited_by_count":2}],"updated_date":"2026-04-04T16:13:02.066488","created_date":"2025-10-10T00:00:00"}
