(original) (raw)

{"id":"https://openalex.org/W2060771462","doi":"https://doi.org/10.1016/j.ejor.2003.04.007","title":"Second order symmetric duality in non-differentiable multiobjective programming with F-convexity","display_name":"Second order symmetric duality in non-differentiable multiobjective programming with F-convexity","publication_year":2004,"publication_date":"2004-02-27","ids":{"openalex":"https://openalex.org/W2060771462","doi":"https://doi.org/10.1016/j.ejor.2003.04.007","mag":"2060771462"},"language":"en","primary\_location":{"id":"doi:10.1016/j.ejor.2003.04.007","is\_oa":false,"landing\_page\_url":"https://doi.org/10.1016/j.ejor.2003.04.007","pdf\_url":null,"source":{"id":"https://openalex.org/S103321696","display\_name":"European Journal of Operational Research","issn_l":"0377-2217","issn":["0377-2217","1872-6860"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320990","host\_organization\_name":"Elsevier BV","host_organization_lineage":["https://openalex.org/P4310320990"\],"host\_organization\_lineage\_names":\["Elsevier BV"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"European Journal of Operational Research","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/A5101830931","display\_name":"Xiaohu Yang","orcid":"https://orcid.org/0000-0001-7427-9751"},"institutions":\[{"id":"https://openalex.org/I126924076","display\_name":"Chongqing Normal University","ror":"https://ror.org/01dcw5w74","country\_code":"CN","type":"education","lineage":\["https://openalex.org/I126924076"\]},{"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":\["CN","HK"\],"is\_corresponding":true,"raw\_author\_name":"X.M. Yang","raw_affiliation_strings":["Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong","Department of Mathematics, Chongqing Normal University, Chongqing 400047, China","Department of Mathematics, Chongqing Normal University, Chongqing 400047, China)"],"affiliations":[{"raw_affiliation_string":"Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong","institution_ids":["https://openalex.org/I14243506"\]},{"raw\_affiliation\_string":"Department of Mathematics, Chongqing Normal University, Chongqing 400047, China","institution_ids":["https://openalex.org/I126924076"\]},{"raw\_affiliation\_string":"Department of Mathematics, Chongqing Normal University, Chongqing 400047, China)","institution_ids":["https://openalex.org/I126924076"\]}\]},{"author\_position":"middle","author":{"id":"https://openalex.org/A5101830931","display\_name":"Xiaohu Yang","orcid":"https://orcid.org/0000-0001-7427-9751"},"institutions":\[{"id":"https://openalex.org/I126924076","display\_name":"Chongqing Normal University","ror":"https://ror.org/01dcw5w74","country\_code":"CN","type":"education","lineage":\["https://openalex.org/I126924076"\]},{"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":\["CN","HK"\],"is\_corresponding":false,"raw\_author\_name":"X.M. Yang","raw_affiliation_strings":["Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong","Department of Mathematics, Chongqing Normal University, Chongqing 400047, China","Department of Applied Mathematics, The Hong Kong Polytechnic University Hung Hom, Kowloon, Hong Kong"],"affiliations":[{"raw_affiliation_string":"Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong","institution_ids":["https://openalex.org/I14243506"\]},{"raw\_affiliation\_string":"Department of Mathematics, Chongqing Normal University, Chongqing 400047, China","institution_ids":["https://openalex.org/I126924076"\]},{"raw\_affiliation\_string":"Department of Applied Mathematics, The Hong Kong Polytechnic University Hung Hom, Kowloon, Hong Kong","institution_ids":["https://openalex.org/I14243506"\]}\]},{"author\_position":"middle","author":{"id":null,"display\_name":"X.Q. Yang","orcid":null},"institutions":[{"id":"https://openalex.org/I126924076","display\_name":"Chongqing Normal University","ror":"https://ror.org/01dcw5w74","country\_code":"CN","type":"education","lineage":\["https://openalex.org/I126924076"\]},{"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":\["CN","HK"\],"is\_corresponding":false,"raw\_author\_name":"X.Q. Yang","raw_affiliation_strings":["Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong","Department of Mathematics, Chongqing Normal University, Chongqing 400047, China","Department of Applied Mathematics, The Hong Kong Polytechnic University Hung Hom, Kowloon, Hong Kong"],"affiliations":[{"raw_affiliation_string":"Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong","institution_ids":["https://openalex.org/I14243506"\]},{"raw\_affiliation\_string":"Department of Mathematics, Chongqing Normal University, Chongqing 400047, China","institution_ids":["https://openalex.org/I126924076"\]},{"raw\_affiliation\_string":"Department of Applied Mathematics, The Hong Kong Polytechnic University Hung Hom, Kowloon, Hong Kong","institution_ids":["https://openalex.org/I14243506"\]}\]},{"author\_position":"middle","author":{"id":null,"display\_name":"X.Q. Yang","orcid":null},"institutions":[{"id":"https://openalex.org/I126924076","display\_name":"Chongqing Normal University","ror":"https://ror.org/01dcw5w74","country\_code":"CN","type":"education","lineage":\["https://openalex.org/I126924076"\]},{"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":\["CN","HK"\],"is\_corresponding":true,"raw\_author\_name":"X.Q. Yang","raw_affiliation_strings":["Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong","Department of Mathematics, Chongqing Normal University, Chongqing 400047, China","Department of Applied Mathematics, The Hong Kong Polytechnic University Hung Hom, Kowloon, Hong Kong"],"affiliations":[{"raw_affiliation_string":"Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong","institution_ids":["https://openalex.org/I14243506"\]},{"raw\_affiliation\_string":"Department of Mathematics, Chongqing Normal University, Chongqing 400047, China","institution_ids":["https://openalex.org/I126924076"\]},{"raw\_affiliation\_string":"Department of Applied Mathematics, The Hong Kong Polytechnic University Hung Hom, Kowloon, Hong Kong","institution_ids":["https://openalex.org/I14243506"\]}\]},{"author\_position":"middle","author":{"id":"https://openalex.org/A5010897184","display\_name":"Kok Lay Teo","orcid":"https://orcid.org/0000-0002-5903-7698"},"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":false,"raw\_author\_name":"K.L. Teo","raw_affiliation_strings":["Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong"],"affiliations":[{"raw_affiliation_string":"Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong","institution_ids":["https://openalex.org/I14243506"\]}\]},{"author\_position":"last","author":{"id":"https://openalex.org/A5085695339","display\_name":"S. H. Hou","orcid":"https://orcid.org/0000-0003-3537-9105"},"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":false,"raw\_author\_name":"S.H. Hou","raw_affiliation_strings":["Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong"],"affiliations":[{"raw_affiliation_string":"Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong","institution_ids":["https://openalex.org/I14243506"\]}\]}\],"institutions":\[\],"countries\_distinct\_count":2,"institutions\_distinct\_count":6,"corresponding\_author\_ids":\["https://openalex.org/A5101830931"\],"corresponding\_institution\_ids":\["https://openalex.org/I126924076","https://openalex.org/I14243506"\],"apc\_list":{"value":3290,"currency":"USD","value\_usd":3290},"apc\_paid":null,"fwci":1.7285,"has\_fulltext":false,"cited\_by\_count":34,"citation\_normalized\_percentile":{"value":0.84261853,"is\_in\_top\_1\_percent":false,"is\_in\_top\_10\_percent":false},"cited\_by\_percentile\_year":{"min":90,"max":98},"biblio":{"volume":"164","issue":"2","first\_page":"406","last\_page":"416"},"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/T11777","display\_name":"Fixed Point Theorems Analysis","score":0.9520999789237976,"subfield":{"id":"https://openalex.org/subfields/2608","display\_name":"Geometry and Topology"},"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/T12709","display\_name":"Optimization and Mathematical Programming","score":0.9251000285148621,"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/duality","display\_name":"Duality (order theory)","score":0.807569146156311},{"id":"https://openalex.org/keywords/convexity","display\_name":"Convexity","score":0.7997435331344604},{"id":"https://openalex.org/keywords/differentiable-function","display\_name":"Differentiable function","score":0.7412911653518677},{"id":"https://openalex.org/keywords/wolfe-duality","display\_name":"Wolfe duality","score":0.6561653017997742},{"id":"https://openalex.org/keywords/mathematics","display\_name":"Mathematics","score":0.6257494688034058},{"id":"https://openalex.org/keywords/order","display\_name":"Order (exchange)","score":0.5799741148948669},{"id":"https://openalex.org/keywords/duality-gap","display\_name":"Duality gap","score":0.5645964741706848},{"id":"https://openalex.org/keywords/dual","display\_name":"Dual (grammatical number)","score":0.5223439931869507},{"id":"https://openalex.org/keywords/strong-duality","display\_name":"Strong duality","score":0.48496174812316895},{"id":"https://openalex.org/keywords/multiobjective-programming","display\_name":"Multiobjective programming","score":0.4263421297073364},{"id":"https://openalex.org/keywords/applied-mathematics","display\_name":"Applied mathematics","score":0.4080798029899597},{"id":"https://openalex.org/keywords/mathematical-optimization","display\_name":"Mathematical optimization","score":0.3893807828426361},{"id":"https://openalex.org/keywords/weak-duality","display\_name":"Weak duality","score":0.3861359655857086},{"id":"https://openalex.org/keywords/mathematical-economics","display\_name":"Mathematical economics","score":0.3648509383201599},{"id":"https://openalex.org/keywords/combinatorics","display\_name":"Combinatorics","score":0.3113602101802826},{"id":"https://openalex.org/keywords/pure-mathematics","display\_name":"Pure mathematics","score":0.30391398072242737},{"id":"https://openalex.org/keywords/optimization-problem","display\_name":"Optimization problem","score":0.1755620837211609},{"id":"https://openalex.org/keywords/multi-objective-optimization","display\_name":"Multi-objective optimization","score":0.15254783630371094},{"id":"https://openalex.org/keywords/economics","display\_name":"Economics","score":0.07803881168365479},{"id":"https://openalex.org/keywords/finance","display\_name":"Finance","score":0.05985832214355469}\],"concepts":\[{"id":"https://openalex.org/C2778023678","wikidata":"https://www.wikidata.org/wiki/Q554403","display\_name":"Duality (order theory)","level":2,"score":0.807569146156311},{"id":"https://openalex.org/C72134830","wikidata":"https://www.wikidata.org/wiki/Q5166524","display\_name":"Convexity","level":2,"score":0.7997435331344604},{"id":"https://openalex.org/C202615002","wikidata":"https://www.wikidata.org/wiki/Q783507","display\_name":"Differentiable function","level":2,"score":0.7412911653518677},{"id":"https://openalex.org/C109456813","wikidata":"https://www.wikidata.org/wiki/Q8029991","display\_name":"Wolfe duality","level":5,"score":0.6561653017997742},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display\_name":"Mathematics","level":0,"score":0.6257494688034058},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display\_name":"Order (exchange)","level":2,"score":0.5799741148948669},{"id":"https://openalex.org/C5274546","wikidata":"https://www.wikidata.org/wiki/Q5310264","display\_name":"Duality gap","level":3,"score":0.5645964741706848},{"id":"https://openalex.org/C2780980858","wikidata":"https://www.wikidata.org/wiki/Q110022","display\_name":"Dual (grammatical number)","level":2,"score":0.5223439931869507},{"id":"https://openalex.org/C163863214","wikidata":"https://www.wikidata.org/wiki/Q7624553","display\_name":"Strong duality","level":3,"score":0.48496174812316895},{"id":"https://openalex.org/C2988952207","wikidata":"https://www.wikidata.org/wiki/Q2052203","display\_name":"Multiobjective programming","level":3,"score":0.4263421297073364},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display\_name":"Applied mathematics","level":1,"score":0.4080798029899597},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display\_name":"Mathematical optimization","level":1,"score":0.3893807828426361},{"id":"https://openalex.org/C80682859","wikidata":"https://www.wikidata.org/wiki/Q7977943","display\_name":"Weak duality","level":4,"score":0.3861359655857086},{"id":"https://openalex.org/C144237770","wikidata":"https://www.wikidata.org/wiki/Q747534","display\_name":"Mathematical economics","level":1,"score":0.3648509383201599},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display\_name":"Combinatorics","level":1,"score":0.3113602101802826},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display\_name":"Pure mathematics","level":1,"score":0.30391398072242737},{"id":"https://openalex.org/C137836250","wikidata":"https://www.wikidata.org/wiki/Q984063","display\_name":"Optimization problem","level":2,"score":0.1755620837211609},{"id":"https://openalex.org/C68781425","wikidata":"https://www.wikidata.org/wiki/Q2052203","display\_name":"Multi-objective optimization","level":2,"score":0.15254783630371094},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display\_name":"Economics","level":0,"score":0.07803881168365479},{"id":"https://openalex.org/C10138342","wikidata":"https://www.wikidata.org/wiki/Q43015","display\_name":"Finance","level":1,"score":0.05985832214355469},{"id":"https://openalex.org/C142362112","wikidata":"https://www.wikidata.org/wiki/Q735","display\_name":"Art","level":0,"score":0.0},{"id":"https://openalex.org/C124952713","wikidata":"https://www.wikidata.org/wiki/Q8242","display\_name":"Literature","level":1,"score":0.0}\],"mesh":\[\],"locations\_count":3,"locations":\[{"id":"doi:10.1016/j.ejor.2003.04.007","is\_oa":false,"landing\_page\_url":"https://doi.org/10.1016/j.ejor.2003.04.007","pdf\_url":null,"source":{"id":"https://openalex.org/S103321696","display\_name":"European Journal of Operational Research","issn_l":"0377-2217","issn":["0377-2217","1872-6860"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320990","host\_organization\_name":"Elsevier BV","host_organization_lineage":["https://openalex.org/P4310320990"\],"host\_organization\_lineage\_names":\["Elsevier BV"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"European Journal of Operational Research","raw_type":"journal-article"},{"id":"pmh:oai:RePEc:eee:ejores:v:164:y:2005:i:2:p:406-416","is_oa":false,"landing_page_url":"http://www.sciencedirect.com/science/article/pii/S0377-2217(04)00027-X","pdf\_url":null,"source":{"id":"https://openalex.org/S4306401271","display\_name":"RePEc: Research Papers in Economics","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I77793887","host\_organization\_name":"Federal Reserve Bank of St. Louis","host_organization_lineage":["https://openalex.org/I77793887"\],"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:10397/14041","is\_oa":false,"landing\_page\_url":"http://hdl.handle.net/10397/14041","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":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":17,"referenced_works":["https://openalex.org/W1979028018","https://openalex.org/W1982946250","https://openalex.org/W1983749636","https://openalex.org/W2000983337","https://openalex.org/W2043250543","https://openalex.org/W2049168249","https://openalex.org/W2064301284","https://openalex.org/W2066080463","https://openalex.org/W2075912492","https://openalex.org/W2088942587","https://openalex.org/W2127489714","https://openalex.org/W2159961705","https://openalex.org/W2160880788","https://openalex.org/W2162753437","https://openalex.org/W2321984564","https://openalex.org/W2564048428","https://openalex.org/W3130212332"\],"related\_works":\["https://openalex.org/W4220928172","https://openalex.org/W2056178476","https://openalex.org/W3213165575","https://openalex.org/W2029896594","https://openalex.org/W4210245775","https://openalex.org/W4283709143","https://openalex.org/W2136200239","https://openalex.org/W2406638882","https://openalex.org/W2040864114","https://openalex.org/W2390707064"\],"abstract\_inverted\_index":null,"counts\_by\_year":\[{"year":2023,"cited\_by\_count":1},{"year":2018,"cited\_by\_count":2},{"year":2017,"cited\_by\_count":1},{"year":2016,"cited\_by\_count":1},{"year":2015,"cited\_by\_count":1},{"year":2014,"cited\_by\_count":1},{"year":2013,"cited\_by\_count":3},{"year":2012,"cited\_by\_count":4}\],"updated\_date":"2026-04-14T08:04:32.555800","created\_date":"2025-10-10T00:00:00"}