{"id":"https://openalex.org/W131053668","doi":"https://doi.org/10.1080/00207160801989867","title":"Generalized ( ,\u03b1, \u03c1, \u03b8)\u2013<i>d</i>\u2013<i>V</i>\u2013univex functions and non-smooth alternative theorems","display_name":"Generalized ( ,\u03b1, \u03c1, \u03b8)\u2013<i>d</i>\u2013<i>V</i>\u2013univex functions and non-smooth alternative theorems","publication_year":2008,"publication_date":"2008-08-21","ids":{"openalex":"https://openalex.org/W131053668","doi":"https://doi.org/10.1080/00207160801989867","mag":"131053668"},"language":"en","primary_location":{"id":"doi:10.1080/00207160801989867","is_oa":false,"landing_page_url":"https://doi.org/10.1080/00207160801989867","pdf_url":null,"source":{"id":"https://openalex.org/S124867444","display_name":"International Journal of Computer Mathematics","issn_l":"0020-7160","issn":["0020-7160","1026-7425","1029-0265"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320547","host_organization_name":"Taylor & Francis","host_organization_lineage":["https://openalex.org/P4310320547"],"host_organization_lineage_names":["Taylor & Francis"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Computer Mathematics","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":null,"display_name":"Zishuang Tong","orcid":null},"institutions":[{"id":"https://openalex.org/I135237710","display_name":"Zhejiang Normal University","ror":"https://ror.org/01vevwk45","country_code":"CN","type":"education","lineage":["https://openalex.org/I135237710"]},{"id":"https://openalex.org/I4210152737","display_name":"Jinhua University of Vocational Technology","ror":"https://ror.org/047hbb113","country_code":"CN","type":"education","lineage":["https://openalex.org/I4210152737"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Zishuang Tong","raw_affiliation_strings":["Normal School of Jinhua College of Profession and Technology, Jinhua, Zhejiang, P. R. China","Normal School of Jinhua College of Profession and Technology, Jinhua, Zhejiang, P. R. China#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Normal School of Jinhua College of Profession and Technology, Jinhua, Zhejiang, P. R. China","institution_ids":["https://openalex.org/I135237710","https://openalex.org/I4210152737"]},{"raw_affiliation_string":"Normal School of Jinhua College of Profession and Technology, Jinhua, Zhejiang, P. R. China#TAB#","institution_ids":["https://openalex.org/I4210152737"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5074913542","display_name":"X. J. Zheng","orcid":null},"institutions":[{"id":"https://openalex.org/I113940042","display_name":"Shanghai University","ror":"https://ror.org/006teas31","country_code":"CN","type":"education","lineage":["https://openalex.org/I113940042"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"X. J. Zheng","raw_affiliation_strings":["Department of Mathematics, Shanghai University, Baoshan, Shanghai, P. R. China","Department of Mathematics, Shanghai University, Shanghai, P. R. China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Shanghai University, Baoshan, Shanghai, P. R. China","institution_ids":["https://openalex.org/I113940042"]},{"raw_affiliation_string":"Department of Mathematics, Shanghai University, Shanghai, P. R. China","institution_ids":["https://openalex.org/I113940042"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I135237710","https://openalex.org/I4210152737"],"apc_list":null,"apc_paid":null,"fwci":0.3937,"has_fulltext":false,"cited_by_count":2,"citation_normalized_percentile":{"value":0.64509967,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"87","issue":"1","first_page":"158","last_page":"172"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10545","display_name":"Optimization and Variational Analysis","score":0.9998999834060669,"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":0.9998999834060669,"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":0.9939000010490417,"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/T11777","display_name":"Fixed Point Theorems Analysis","score":0.9894999861717224,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8537901639938354},{"id":"https://openalex.org/keywords/convexity","display_name":"Convexity","score":0.7401114702224731},{"id":"https://openalex.org/keywords/corollary","display_name":"Corollary","score":0.6937710642814636},{"id":"https://openalex.org/keywords/saddle-point","display_name":"Saddle point","score":0.6175806522369385},{"id":"https://openalex.org/keywords/regular-polygon","display_name":"Regular polygon","score":0.5007748603820801},{"id":"https://openalex.org/keywords/saddle","display_name":"Saddle","score":0.4836055338382721},{"id":"https://openalex.org/keywords/convex-function","display_name":"Convex function","score":0.45885637402534485},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.43915510177612305},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.43904826045036316},{"id":"https://openalex.org/keywords/type","display_name":"Type (biology)","score":0.4227566719055176},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.2348039448261261}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8537901639938354},{"id":"https://openalex.org/C72134830","wikidata":"https://www.wikidata.org/wiki/Q5166524","display_name":"Convexity","level":2,"score":0.7401114702224731},{"id":"https://openalex.org/C2780012671","wikidata":"https://www.wikidata.org/wiki/Q1343870","display_name":"Corollary","level":2,"score":0.6937710642814636},{"id":"https://openalex.org/C2681867","wikidata":"https://www.wikidata.org/wiki/Q690935","display_name":"Saddle point","level":2,"score":0.6175806522369385},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.5007748603820801},{"id":"https://openalex.org/C2777127463","wikidata":"https://www.wikidata.org/wiki/Q10862618","display_name":"Saddle","level":2,"score":0.4836055338382721},{"id":"https://openalex.org/C145446738","wikidata":"https://www.wikidata.org/wiki/Q319913","display_name":"Convex function","level":3,"score":0.45885637402534485},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.43915510177612305},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.43904826045036316},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.4227566719055176},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.2348039448261261},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C18903297","wikidata":"https://www.wikidata.org/wiki/Q7150","display_name":"Ecology","level":1,"score":0.0},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0},{"id":"https://openalex.org/C106159729","wikidata":"https://www.wikidata.org/wiki/Q2294553","display_name":"Financial economics","level":1,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1080/00207160801989867","is_oa":false,"landing_page_url":"https://doi.org/10.1080/00207160801989867","pdf_url":null,"source":{"id":"https://openalex.org/S124867444","display_name":"International Journal of Computer Mathematics","issn_l":"0020-7160","issn":["0020-7160","1026-7425","1029-0265"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320547","host_organization_name":"Taylor & Francis","host_organization_lineage":["https://openalex.org/P4310320547"],"host_organization_lineage_names":["Taylor & Francis"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Computer Mathematics","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":23,"referenced_works":["https://openalex.org/W143702628","https://openalex.org/W152156553","https://openalex.org/W184567933","https://openalex.org/W187778025","https://openalex.org/W1501511935","https://openalex.org/W1551360398","https://openalex.org/W1590012787","https://openalex.org/W1984812465","https://openalex.org/W1991014520","https://openalex.org/W2000983337","https://openalex.org/W2008688745","https://openalex.org/W2027302763","https://openalex.org/W2040167656","https://openalex.org/W2044088357","https://openalex.org/W2079479228","https://openalex.org/W2094300862","https://openalex.org/W2103732289","https://openalex.org/W2153015072","https://openalex.org/W2313488898","https://openalex.org/W2317002898","https://openalex.org/W2321984564","https://openalex.org/W4249667877","https://openalex.org/W4256495659"],"related_works":["https://openalex.org/W4377200937","https://openalex.org/W4236459141","https://openalex.org/W4205304778","https://openalex.org/W2020252434","https://openalex.org/W73248859","https://openalex.org/W2584253892","https://openalex.org/W2350324449","https://openalex.org/W119381072","https://openalex.org/W2034033896","https://openalex.org/W2094891943"],"abstract_inverted_index":{"Under":[0],"a":[1,61,67],"new":[2],"kind":[3],"of":[4,38,48],"generalized":[5,9,27],"convexity":[6],"and":[7,20],"non-smooth":[8],"(,":[10],"\u03b1,":[11],"\u03c1,":[12],"\u03b8)\u2013d\u2013V\u2013univexity,":[13],"some":[14,25],"basic":[15],"alternative":[16,29,55,64],"theorems":[17,56],"are":[18,57],"established":[19],"proved.":[21],"These":[22],"results":[23],"extend":[24],"other":[26],"convex":[28,63],"theorems.":[30],"To":[31,52],"illustrate":[32],"their":[33],"applications,":[34],"we":[35],"present":[36],"characterizations":[37],"saddle-point":[39],"type":[40],"necessary":[41],"optimality":[42],"conditions":[43],"for":[44],"properly":[45],"efficient":[46],"solutions":[47],"vector":[49],"optimization":[50],"problems.":[51],"conclude,":[53],"the":[54],"applied":[58],"to":[59],"yield":[60],"canonical":[62],"theorem":[65],"as":[66],"corollary.":[68]},"counts_by_year":[{"year":2013,"cited_by_count":1}],"updated_date":"2026-07-29T09:40:50.615796","created_date":"2025-10-10T00:00:00"}
