{"id":"https://openalex.org/W2001734490","doi":"https://doi.org/10.1007/s10957-010-9710-5","title":"Structure and Weak Sharp Minimum of the Pareto Solution Set for Piecewise Linear Multiobjective Optimization","display_name":"Structure and Weak Sharp Minimum of the Pareto Solution Set for Piecewise Linear Multiobjective Optimization","publication_year":2010,"publication_date":"2010-05-11","ids":{"openalex":"https://openalex.org/W2001734490","doi":"https://doi.org/10.1007/s10957-010-9710-5","mag":"2001734490"},"language":"en","primary_location":{"id":"doi:10.1007/s10957-010-9710-5","is_oa":false,"landing_page_url":"https://doi.org/10.1007/s10957-010-9710-5","pdf_url":null,"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":null,"license_id":null,"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":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5024523994","display_name":"Xiaoqi Yang","orcid":"https://orcid.org/0000-0002-5583-4032"},"institutions":[{"id":"https://openalex.org/I14243506","display_name":"Hong Kong Polytechnic University","ror":"https://ror.org/0030zas98","country_code":"HK","type":"education","lineage":["https://openalex.org/I14243506"]}],"countries":["HK"],"is_corresponding":true,"raw_author_name":"X. Q. Yang","raw_affiliation_strings":["Department of Applied Mathematics, Hong Kong Polytechnic University, Hung Hom, Hong Kong","Hong Kong Polytechnic University"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Applied Mathematics, Hong Kong Polytechnic University, Hung Hom, Hong Kong","institution_ids":["https://openalex.org/I14243506"]},{"raw_affiliation_string":"Hong Kong Polytechnic University","institution_ids":["https://openalex.org/I14243506"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5061405938","display_name":"N. D. Yen","orcid":"https://orcid.org/0000-0001-7250-7933"},"institutions":[{"id":"https://openalex.org/I4210148301","display_name":"Institute of Mathematics","ror":"https://ror.org/04jr71r83","country_code":"VN","type":"facility","lineage":["https://openalex.org/I4210148301","https://openalex.org/I70349855"]},{"id":"https://openalex.org/I70349855","display_name":"Vietnam Academy of Science and Technology","ror":"https://ror.org/02wsd5p50","country_code":"VN","type":"government","lineage":["https://openalex.org/I70349855"]}],"countries":["VN"],"is_corresponding":false,"raw_author_name":"N. D. Yen","raw_affiliation_strings":["Institute of Mathematics, Vietnamese Academy of Science and Technology, Hanoi, Vietnam","Vietnamese Academy of Science and Technology"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute of Mathematics, Vietnamese Academy of Science and Technology, Hanoi, Vietnam","institution_ids":["https://openalex.org/I4210148301","https://openalex.org/I70349855"]},{"raw_affiliation_string":"Vietnamese Academy of Science and Technology","institution_ids":["https://openalex.org/I70349855"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5024523994"],"corresponding_institution_ids":["https://openalex.org/I14243506"],"apc_list":{"value":2990,"currency":"USD","value_usd":2990},"apc_paid":null,"fwci":2.2264,"has_fulltext":false,"cited_by_count":26,"citation_normalized_percentile":{"value":0.86984336,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"147","issue":"1","first_page":"113","last_page":"124"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10545","display_name":"Optimization and Variational Analysis","score":0.9998000264167786,"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":0.9998000264167786,"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":0.9984999895095825,"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/T10848","display_name":"Advanced Multi-Objective Optimization Algorithms","score":0.9915000200271606,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8440589904785156},{"id":"https://openalex.org/keywords/theory-of-computation","display_name":"Theory of computation","score":0.6759294271469116},{"id":"https://openalex.org/keywords/polyhedron","display_name":"Polyhedron","score":0.6451585292816162},{"id":"https://openalex.org/keywords/piecewise","display_name":"Piecewise","score":0.5517131090164185},{"id":"https://openalex.org/keywords/piecewise-linear-function","display_name":"Piecewise linear function","score":0.5432078242301941},{"id":"https://openalex.org/keywords/regular-polygon","display_name":"Regular polygon","score":0.5276600122451782},{"id":"https://openalex.org/keywords/pareto-principle","display_name":"Pareto principle","score":0.5171725153923035},{"id":"https://openalex.org/keywords/cone","display_name":"Cone (formal languages)","score":0.5030128359794617},{"id":"https://openalex.org/keywords/space","display_name":"Space (punctuation)","score":0.48258745670318604},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.4752059876918793},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.4747924208641052},{"id":"https://openalex.org/keywords/vector-optimization","display_name":"Vector optimization","score":0.4672413170337677},{"id":"https://openalex.org/keywords/solution-set","display_name":"Solution set","score":0.446399986743927},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.3825696110725403},{"id":"https://openalex.org/keywords/optimization-problem","display_name":"Optimization problem","score":0.347015917301178},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.24455204606056213},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.132980078458786},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.12347033619880676},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.0781095027923584}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8440589904785156},{"id":"https://openalex.org/C24858836","wikidata":"https://www.wikidata.org/wiki/Q844718","display_name":"Theory of computation","level":2,"score":0.6759294271469116},{"id":"https://openalex.org/C54829058","wikidata":"https://www.wikidata.org/wiki/Q172937","display_name":"Polyhedron","level":2,"score":0.6451585292816162},{"id":"https://openalex.org/C164660894","wikidata":"https://www.wikidata.org/wiki/Q2037833","display_name":"Piecewise","level":2,"score":0.5517131090164185},{"id":"https://openalex.org/C17095337","wikidata":"https://www.wikidata.org/wiki/Q2375229","display_name":"Piecewise linear function","level":2,"score":0.5432078242301941},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.5276600122451782},{"id":"https://openalex.org/C137635306","wikidata":"https://www.wikidata.org/wiki/Q182667","display_name":"Pareto principle","level":2,"score":0.5171725153923035},{"id":"https://openalex.org/C30014739","wikidata":"https://www.wikidata.org/wiki/Q5159445","display_name":"Cone (formal languages)","level":2,"score":0.5030128359794617},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.48258745670318604},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.4752059876918793},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.4747924208641052},{"id":"https://openalex.org/C164088818","wikidata":"https://www.wikidata.org/wiki/Q7917833","display_name":"Vector optimization","level":4,"score":0.4672413170337677},{"id":"https://openalex.org/C60033838","wikidata":"https://www.wikidata.org/wiki/Q1751685","display_name":"Solution set","level":3,"score":0.446399986743927},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3825696110725403},{"id":"https://openalex.org/C137836250","wikidata":"https://www.wikidata.org/wiki/Q984063","display_name":"Optimization problem","level":2,"score":0.347015917301178},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.24455204606056213},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.132980078458786},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.12347033619880676},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.0781095027923584},{"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/C111919701","wikidata":"https://www.wikidata.org/wiki/Q9135","display_name":"Operating system","level":1,"score":0.0},{"id":"https://openalex.org/C122357587","wikidata":"https://www.wikidata.org/wiki/Q6934508","display_name":"Multi-swarm optimization","level":3,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1007/s10957-010-9710-5","is_oa":false,"landing_page_url":"https://doi.org/10.1007/s10957-010-9710-5","pdf_url":null,"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":null,"license_id":null,"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:147:y:2010:i:1:d:10.1007_s10957-010-9710-5","is_oa":false,"landing_page_url":"http://link.springer.com/10.1007/s10957-010-9710-5","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"},{"id":"pmh:oai:ira.lib.polyu.edu.hk:10397/32723","is_oa":false,"landing_page_url":"http://hdl.handle.net/10397/32723","pdf_url":null,"source":{"id":"https://openalex.org/S4306400205","display_name":"PolyU Institutional Research Archive (Hong Kong Polytechnic University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I14243506","host_organization_name":"Hong Kong Polytechnic University","host_organization_lineage":["https://openalex.org/I14243506"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Journal/Magazine Article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":22,"referenced_works":["https://openalex.org/W113278869","https://openalex.org/W1538724032","https://openalex.org/W1971559451","https://openalex.org/W1978090048","https://openalex.org/W1982248867","https://openalex.org/W1982422114","https://openalex.org/W1994803283","https://openalex.org/W2008653424","https://openalex.org/W2035136005","https://openalex.org/W2038377767","https://openalex.org/W2040217797","https://openalex.org/W2057428699","https://openalex.org/W2061698680","https://openalex.org/W2064114733","https://openalex.org/W2075272819","https://openalex.org/W2506044023","https://openalex.org/W2561618607","https://openalex.org/W3217569503","https://openalex.org/W4235851924","https://openalex.org/W4249125060","https://openalex.org/W4249513058","https://openalex.org/W4249667877"],"related_works":["https://openalex.org/W2138511311","https://openalex.org/W1994436307","https://openalex.org/W3183145980","https://openalex.org/W1483856296","https://openalex.org/W2068168877","https://openalex.org/W2292688905","https://openalex.org/W2329573458","https://openalex.org/W3107809632","https://openalex.org/W2074614164","https://openalex.org/W2240218224"],"abstract_inverted_index":null,"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":1},{"year":2022,"cited_by_count":2},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":5},{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":4},{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":1},{"year":2013,"cited_by_count":1},{"year":2012,"cited_by_count":3}],"updated_date":"2026-07-23T08:03:31.855105","created_date":"2025-10-10T00:00:00"}