{"id":"https://openalex.org/W4382403329","doi":"https://doi.org/10.3390/axioms12070635","title":"Optimality Conditions of the Approximate Efficiency for Nonsmooth Robust Multiobjective Fractional Semi-Infinite Optimization Problems","display_name":"Optimality Conditions of the Approximate Efficiency for Nonsmooth Robust Multiobjective Fractional Semi-Infinite Optimization Problems","publication_year":2023,"publication_date":"2023-06-27","ids":{"openalex":"https://openalex.org/W4382403329","doi":"https://doi.org/10.3390/axioms12070635"},"language":"en","primary_location":{"id":"doi:10.3390/axioms12070635","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms12070635","pdf_url":"https://www.mdpi.com/2075-1680/12/7/635/pdf?version=1687939941","source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Axioms","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2075-1680/12/7/635/pdf?version=1687939941","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5103673266","display_name":"Liu Gao","orcid":null},"institutions":[{"id":"https://openalex.org/I3019309368","display_name":"North Minzu University","ror":"https://ror.org/05xjevr11","country_code":"CN","type":"education","lineage":["https://openalex.org/I3019309368"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Liu Gao","raw_affiliation_strings":["School of Mathematics and Information Science, North Minzu University, Yinchuan 750021, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematics and Information Science, North Minzu University, Yinchuan 750021, China","institution_ids":["https://openalex.org/I3019309368"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5076538595","display_name":"Guolin Yu","orcid":"https://orcid.org/0000-0003-4729-7748"},"institutions":[{"id":"https://openalex.org/I3019309368","display_name":"North Minzu University","ror":"https://ror.org/05xjevr11","country_code":"CN","type":"education","lineage":["https://openalex.org/I3019309368"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Guolin Yu","raw_affiliation_strings":["School of Mathematics and Information Science, North Minzu University, Yinchuan 750021, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematics and Information Science, North Minzu University, Yinchuan 750021, China","institution_ids":["https://openalex.org/I3019309368"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5029130423","display_name":"Wenyan Han","orcid":null},"institutions":[{"id":"https://openalex.org/I21642278","display_name":"Ningxia University","ror":"https://ror.org/04j7b2v61","country_code":"CN","type":"education","lineage":["https://openalex.org/I21642278"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Wenyan Han","raw_affiliation_strings":["School of Mathematics and Statistics, Ningxia University, Yinchuan 750021, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematics and Statistics, Ningxia University, Yinchuan 750021, China","institution_ids":["https://openalex.org/I21642278"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5076538595"],"corresponding_institution_ids":["https://openalex.org/I3019309368"],"apc_list":{"value":1400,"currency":"CHF","value_usd":1515},"apc_paid":{"value":1400,"currency":"CHF","value_usd":1515},"fwci":0.3743,"has_fulltext":true,"cited_by_count":2,"citation_normalized_percentile":{"value":0.63246952,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":95},"biblio":{"volume":"12","issue":"7","first_page":"635","last_page":"635"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10545","display_name":"Optimization and Variational Analysis","score":0.9995999932289124,"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":0.9995999932289124,"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/T11413","display_name":"Risk and Portfolio Optimization","score":0.9897000193595886,"subfield":{"id":"https://openalex.org/subfields/1803","display_name":"Management Science and Operations Research"},"field":{"id":"https://openalex.org/fields/18","display_name":"Decision Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},{"id":"https://openalex.org/T10969","display_name":"Water resources management and optimization","score":0.989300012588501,"subfield":{"id":"https://openalex.org/subfields/2212","display_name":"Ocean 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/saddle-point","display_name":"Saddle point","score":0.8393044471740723},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7709087133407593},{"id":"https://openalex.org/keywords/saddle","display_name":"Saddle","score":0.6160889863967896},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.5466935038566589},{"id":"https://openalex.org/keywords/regular-polygon","display_name":"Regular polygon","score":0.5157904624938965},{"id":"https://openalex.org/keywords/point","display_name":"Point (geometry)","score":0.49052852392196655},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4679511785507202},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.4613265097141266},{"id":"https://openalex.org/keywords/optimization-problem","display_name":"Optimization problem","score":0.44401097297668457},{"id":"https://openalex.org/keywords/convex-function","display_name":"Convex function","score":0.43638816475868225},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.058264583349227905}],"concepts":[{"id":"https://openalex.org/C2681867","wikidata":"https://www.wikidata.org/wiki/Q690935","display_name":"Saddle point","level":2,"score":0.8393044471740723},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7709087133407593},{"id":"https://openalex.org/C2777127463","wikidata":"https://www.wikidata.org/wiki/Q10862618","display_name":"Saddle","level":2,"score":0.6160889863967896},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.5466935038566589},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.5157904624938965},{"id":"https://openalex.org/C28719098","wikidata":"https://www.wikidata.org/wiki/Q44946","display_name":"Point (geometry)","level":2,"score":0.49052852392196655},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4679511785507202},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.4613265097141266},{"id":"https://openalex.org/C137836250","wikidata":"https://www.wikidata.org/wiki/Q984063","display_name":"Optimization problem","level":2,"score":0.44401097297668457},{"id":"https://openalex.org/C145446738","wikidata":"https://www.wikidata.org/wiki/Q319913","display_name":"Convex function","level":3,"score":0.43638816475868225},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.058264583349227905},{"id":"https://openalex.org/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","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:10.3390/axioms12070635","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms12070635","pdf_url":"https://www.mdpi.com/2075-1680/12/7/635/pdf?version=1687939941","source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Axioms","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:bfbf26ae02594b98b0f3ad2474d94e1b","is_oa":true,"landing_page_url":"https://doaj.org/article/bfbf26ae02594b98b0f3ad2474d94e1b","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Axioms, Vol 12, Iss 7, p 635 (2023)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2075-1680/12/7/635/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/axioms12070635","pdf_url":null,"source":{"id":"https://openalex.org/S4306400947","display_name":"MDPI (MDPI AG)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210097602","host_organization_name":"Multidisciplinary Digital Publishing Institute (Switzerland)","host_organization_lineage":["https://openalex.org/I4210097602"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Axioms; Volume 12; Issue 7; Pages: 635","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/axioms12070635","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms12070635","pdf_url":"https://www.mdpi.com/2075-1680/12/7/635/pdf?version=1687939941","source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Axioms","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G2556025005","display_name":null,"funder_award_id":"2022BSB03046","funder_id":"https://openalex.org/F4320334010","funder_display_name":"Key Research and Development Program of Ningxia"},{"id":"https://openalex.org/G5171080036","display_name":"\u5411\u91cf\u5747\u8861\u95ee\u9898\u89e3\u7684\u6700\u4f18\u6027\u4e0e\u8fde\u7eed\u6027\u7814\u7a76","funder_award_id":"11861002","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G5649835522","display_name":null,"funder_award_id":"2021KYQD23","funder_id":"https://openalex.org/F4320335787","funder_display_name":"Fundamental Research Funds for the Central Universities"},{"id":"https://openalex.org/G7435933562","display_name":null,"funder_award_id":"ZDZX201804","funder_id":"https://openalex.org/F4320327619","funder_display_name":"North Minzu University"},{"id":"https://openalex.org/G7673574296","display_name":null,"funder_award_id":"ZDZX201804","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G8156364870","display_name":null,"funder_award_id":"2022AAC03260","funder_id":"https://openalex.org/F4320322877","funder_display_name":"Natural Science Foundation of Ningxia Province"},{"id":"https://openalex.org/G8936960699","display_name":null,"funder_award_id":"2022XYZSX03","funder_id":"https://openalex.org/F4320335787","funder_display_name":"Fundamental Research Funds for the Central Universities"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320322877","display_name":"Natural Science Foundation of Ningxia Province","ror":null},{"id":"https://openalex.org/F4320327619","display_name":"North Minzu University","ror":null},{"id":"https://openalex.org/F4320334010","display_name":"Key Research and Development Program of Ningxia","ror":null},{"id":"https://openalex.org/F4320335787","display_name":"Fundamental Research Funds for the Central Universities","ror":null}],"has_content":{"grobid_xml":false,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4382403329.pdf"},"referenced_works_count":22,"referenced_works":["https://openalex.org/W44597993","https://openalex.org/W1936391100","https://openalex.org/W2056604947","https://openalex.org/W2094964517","https://openalex.org/W2182144464","https://openalex.org/W2271909439","https://openalex.org/W2324787199","https://openalex.org/W2591170312","https://openalex.org/W2743593635","https://openalex.org/W2766278219","https://openalex.org/W2901345478","https://openalex.org/W3022391784","https://openalex.org/W3044640140","https://openalex.org/W3156761120","https://openalex.org/W3195250998","https://openalex.org/W4224285156","https://openalex.org/W4251616545","https://openalex.org/W4280632251","https://openalex.org/W4319315033","https://openalex.org/W6694046090","https://openalex.org/W6781015436","https://openalex.org/W6838635985"],"related_works":["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","https://openalex.org/W4240399920"],"abstract_inverted_index":{"This":[0],"paper":[1],"is":[2,35,52,66],"devoted":[3],"to":[4,80],"the":[5,39,42,55,71,81],"investigation":[6],"of":[7,41,48,73],"optimality":[8,33,64],"conditions":[9],"and":[10,58,84],"saddle":[11,78,87],"point":[12,79,88],"theorems":[13],"for":[14,20,54],"robust":[15,75],"approximate":[16,49,76],"quasi-weak":[17,77],"efficient":[18],"solutions":[19],"a":[21,31,46,62,74],"nonsmooth":[22],"uncertain":[23],"multiobjective":[24],"fractional":[25],"semi-infinite":[26],"optimization":[27],"problem":[28,56,82],"(NUMFP).":[29],"Firstly,":[30],"necessary":[32],"condition":[34,65],"established":[36],"by":[37],"using":[38],"properties":[40],"Gerstewitz\u2019s":[43],"function.":[44],"Furthermore,":[45],"kind":[47],"pseudo/quasi-convex":[50],"function":[51],"defined":[53],"(NUMFP),":[57],"under":[59],"its":[60],"assumption,":[61],"sufficient":[63],"obtained.":[67],"Finally,":[68],"we":[69],"introduce":[70],"notion":[72],"(NUMFP)":[83],"prove":[85],"corresponding":[86],"theorems.":[89]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":1}],"updated_date":"2026-06-06T09:05:17.133730","created_date":"2023-06-29T00:00:00"}