{"id":"https://openalex.org/W4213385679","doi":"https://doi.org/10.1007/s10700-022-09381-4","title":"Optimality conditions for invex nonsmooth optimization problems with fuzzy objective functions","display_name":"Optimality conditions for invex nonsmooth optimization problems with fuzzy objective functions","publication_year":2022,"publication_date":"2022-02-22","ids":{"openalex":"https://openalex.org/W4213385679","doi":"https://doi.org/10.1007/s10700-022-09381-4"},"language":"en","primary_location":{"id":"doi:10.1007/s10700-022-09381-4","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10700-022-09381-4","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10700-022-09381-4.pdf","source":{"id":"https://openalex.org/S93986401","display_name":"Fuzzy Optimization and Decision Making","issn_l":"1568-4539","issn":["1568-4539","1573-2908"],"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Fuzzy Optimization and Decision Making","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s10700-022-09381-4.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5050602810","display_name":"Tadeusz Antczak","orcid":"https://orcid.org/0000-0002-2218-7668"},"institutions":[{"id":"https://openalex.org/I34250744","display_name":"University of \u0141\u00f3d\u017a","ror":"https://ror.org/05cq64r17","country_code":"PL","type":"education","lineage":["https://openalex.org/I34250744"]}],"countries":["PL"],"is_corresponding":true,"raw_author_name":"Tadeusz Antczak","raw_affiliation_strings":["Faculty of Mathematics and Computer Science, University of \u0141\u00f3d\u017a, Banacha 22, 90-238, \u0141\u00f3d\u017a, Poland"],"raw_orcid":"https://orcid.org/0000-0002-2218-7668","affiliations":[{"raw_affiliation_string":"Faculty of Mathematics and Computer Science, University of \u0141\u00f3d\u017a, Banacha 22, 90-238, \u0141\u00f3d\u017a, Poland","institution_ids":["https://openalex.org/I34250744"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5050602810"],"corresponding_institution_ids":["https://openalex.org/I34250744"],"apc_list":{"value":2590,"currency":"EUR","value_usd":3390},"apc_paid":{"value":2590,"currency":"EUR","value_usd":3390},"fwci":1.7261,"has_fulltext":true,"cited_by_count":8,"citation_normalized_percentile":{"value":0.84663799,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":96,"max":98},"biblio":{"volume":"22","issue":"1","first_page":"1","last_page":"21"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12135","display_name":"Fuzzy Systems and Optimization","score":0.9997000098228455,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T12135","display_name":"Fuzzy Systems and Optimization","score":0.9997000098228455,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"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/T10545","display_name":"Optimization and Variational Analysis","score":0.9993000030517578,"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/T12709","display_name":"Optimization and Mathematical Programming","score":0.9957000017166138,"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":0.7319256067276001},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.6824949979782104},{"id":"https://openalex.org/keywords/weighting","display_name":"Weighting","score":0.6716525554656982},{"id":"https://openalex.org/keywords/lipschitz-continuity","display_name":"Lipschitz continuity","score":0.6048038005828857},{"id":"https://openalex.org/keywords/optimization-problem","display_name":"Optimization problem","score":0.593143105506897},{"id":"https://openalex.org/keywords/fuzzy-logic","display_name":"Fuzzy logic","score":0.5237091183662415},{"id":"https://openalex.org/keywords/multi-objective-optimization","display_name":"Multi-objective optimization","score":0.4714677631855011},{"id":"https://openalex.org/keywords/pareto-principle","display_name":"Pareto principle","score":0.4648175835609436},{"id":"https://openalex.org/keywords/minification","display_name":"Minification","score":0.4646548628807068},{"id":"https://openalex.org/keywords/directional-derivative","display_name":"Directional derivative","score":0.41661468148231506},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.34222325682640076},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.23807331919670105},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.14146071672439575},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.05496677756309509},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.05310368537902832}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7319256067276001},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.6824949979782104},{"id":"https://openalex.org/C183115368","wikidata":"https://www.wikidata.org/wiki/Q856577","display_name":"Weighting","level":2,"score":0.6716525554656982},{"id":"https://openalex.org/C22324862","wikidata":"https://www.wikidata.org/wiki/Q652707","display_name":"Lipschitz continuity","level":2,"score":0.6048038005828857},{"id":"https://openalex.org/C137836250","wikidata":"https://www.wikidata.org/wiki/Q984063","display_name":"Optimization problem","level":2,"score":0.593143105506897},{"id":"https://openalex.org/C58166","wikidata":"https://www.wikidata.org/wiki/Q224821","display_name":"Fuzzy logic","level":2,"score":0.5237091183662415},{"id":"https://openalex.org/C68781425","wikidata":"https://www.wikidata.org/wiki/Q2052203","display_name":"Multi-objective optimization","level":2,"score":0.4714677631855011},{"id":"https://openalex.org/C137635306","wikidata":"https://www.wikidata.org/wiki/Q182667","display_name":"Pareto principle","level":2,"score":0.4648175835609436},{"id":"https://openalex.org/C147764199","wikidata":"https://www.wikidata.org/wiki/Q6865248","display_name":"Minification","level":2,"score":0.4646548628807068},{"id":"https://openalex.org/C2011187","wikidata":"https://www.wikidata.org/wiki/Q383851","display_name":"Directional derivative","level":2,"score":0.41661468148231506},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.34222325682640076},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.23807331919670105},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.14146071672439575},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.05496677756309509},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.05310368537902832},{"id":"https://openalex.org/C126838900","wikidata":"https://www.wikidata.org/wiki/Q77604","display_name":"Radiology","level":1,"score":0.0},{"id":"https://openalex.org/C71924100","wikidata":"https://www.wikidata.org/wiki/Q11190","display_name":"Medicine","level":0,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1007/s10700-022-09381-4","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10700-022-09381-4","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10700-022-09381-4.pdf","source":{"id":"https://openalex.org/S93986401","display_name":"Fuzzy Optimization and Decision Making","issn_l":"1568-4539","issn":["1568-4539","1573-2908"],"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Fuzzy Optimization and Decision Making","raw_type":"journal-article"},{"id":"pmh:oai:RePEc:spr:fuzodm:v:22:y:2023:i:1:d:10.1007_s10700-022-09381-4","is_oa":false,"landing_page_url":"http://link.springer.com/10.1007/s10700-022-09381-4","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"}],"best_oa_location":{"id":"doi:10.1007/s10700-022-09381-4","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10700-022-09381-4","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10700-022-09381-4.pdf","source":{"id":"https://openalex.org/S93986401","display_name":"Fuzzy Optimization and Decision Making","issn_l":"1568-4539","issn":["1568-4539","1573-2908"],"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Fuzzy Optimization and Decision Making","raw_type":"journal-article"},"sustainable_development_goals":[{"display_name":"Reduced inequalities","score":0.41999998688697815,"id":"https://metadata.un.org/sdg/10"}],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4213385679.pdf","grobid_xml":"https://content.openalex.org/works/W4213385679.grobid-xml"},"referenced_works_count":24,"referenced_works":["https://openalex.org/W636069421","https://openalex.org/W1551360398","https://openalex.org/W1978499389","https://openalex.org/W1980528869","https://openalex.org/W1987832787","https://openalex.org/W1996452001","https://openalex.org/W1997903931","https://openalex.org/W2022336732","https://openalex.org/W2037592439","https://openalex.org/W2039064061","https://openalex.org/W2050676121","https://openalex.org/W2052664014","https://openalex.org/W2056215967","https://openalex.org/W2072707988","https://openalex.org/W2078839838","https://openalex.org/W2094172591","https://openalex.org/W2094219041","https://openalex.org/W2094779154","https://openalex.org/W2176210190","https://openalex.org/W4256242347","https://openalex.org/W4256495659","https://openalex.org/W6632629376","https://openalex.org/W6652304802","https://openalex.org/W6652548972"],"related_works":["https://openalex.org/W2090178682","https://openalex.org/W4241467429","https://openalex.org/W2073147994","https://openalex.org/W2001591765","https://openalex.org/W2384474142","https://openalex.org/W3083133203","https://openalex.org/W1588199609","https://openalex.org/W2744462909","https://openalex.org/W1971520370","https://openalex.org/W4243698225"],"abstract_inverted_index":{"Abstract":[0],"In":[1],"this":[2],"paper,":[3],"the":[4,63,70,96,107,128],"definitions":[5],"of":[6,127],"Clarke":[7,18],"generalized":[8,19],"directional":[9],"$$\\alpha":[10],"$$":[11],"<mml:math":[12],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">":[13],"<mml:mi>\u03b1</mml:mi>":[14],"</mml:math>":[15],"-derivative":[16],"and":[17,39,42,124],"gradient":[20],"are":[21,51,113],"introduced":[22],"for":[23,53,69],"a":[24,30,55],"locally":[25],"Lipschitz":[26],"fuzzy":[27,36,73,110],"function.":[28],"Further,":[29],"nonconvex":[31],"nonsmooth":[32,56,72,109],"optimization":[33,74,79,85,122],"problem":[34,80,86,93,112,123],"with":[35],"objective":[37],"function":[38],"both":[40],"inequality":[41],"equality":[43],"constraints":[44],"is":[45,65,81,87],"considered.":[46],"The":[47,83],"Karush-Kuhn-Tucker":[48,125],"optimality":[49],"conditions":[50],"established":[52],"such":[54],"extremum":[57],"problem.":[58,130],"For":[59],"proving":[60],"these":[61],"conditions,":[62],"approach":[64],"used":[66],"in":[67,95,106,118],"which,":[68],"considered":[71,108],"problem,":[75],"its":[76,90,119],"associated":[77,91,120],"bi-objective":[78,84,121],"constructed.":[82],"solved":[88],"by":[89],"scalarized":[92],"constructed":[94],"weighting":[97,129],"method.":[98],"Then,":[99],"under":[100],"invexity":[101],"hypotheses,":[102],"(weakly)":[103],"nondominated":[104],"solutions":[105,117],"minimization":[111],"characterized":[114],"through":[115],"Pareto":[116],"points":[126]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":4},{"year":2024,"cited_by_count":3}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}
