{"id":"https://openalex.org/W1977158646","doi":"https://doi.org/10.1155/ijmms/2006/30459","title":"Duality without constraint qualification in nonsmooth optimization","display_name":"Duality without constraint qualification in nonsmooth optimization","publication_year":2006,"publication_date":"2006-01-01","ids":{"openalex":"https://openalex.org/W1977158646","doi":"https://doi.org/10.1155/ijmms/2006/30459","mag":"1977158646"},"language":"en","primary_location":{"id":"doi:10.1155/ijmms/2006/30459","is_oa":true,"landing_page_url":"https://doi.org/10.1155/ijmms/2006/30459","pdf_url":"https://downloads.hindawi.com/journals/ijmms/2006/030459.pdf","source":{"id":"https://openalex.org/S151851439","display_name":"International Journal of Mathematics and Mathematical Sciences","issn_l":"0161-1712","issn":["0161-1712","1687-0425"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310319869","host_organization_name":"Hindawi Publishing Corporation","host_organization_lineage":["https://openalex.org/P4310319869","https://openalex.org/P4310320595"],"host_organization_lineage_names":["Hindawi Publishing Corporation","Wiley"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Mathematics and Mathematical Sciences","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://downloads.hindawi.com/journals/ijmms/2006/030459.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5060367415","display_name":"S. Nobakhtian","orcid":"https://orcid.org/0000-0003-2488-7892"},"institutions":[{"id":"https://openalex.org/I39268498","display_name":"University of Isfahan","ror":"https://ror.org/05h9t7759","country_code":"IR","type":"education","lineage":["https://openalex.org/I39268498"]}],"countries":["IR"],"is_corresponding":true,"raw_author_name":"S. Nobakhtian","raw_affiliation_strings":["Department of Mathematics, University of Isfahan, Isfahan 81745-163, Iran"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Isfahan, Isfahan 81745-163, Iran","institution_ids":["https://openalex.org/I39268498"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5060367415"],"corresponding_institution_ids":["https://openalex.org/I39268498"],"apc_list":{"value":1050,"currency":"USD","value_usd":1050},"apc_paid":{"value":1050,"currency":"USD","value_usd":1050},"fwci":0.3247,"has_fulltext":true,"cited_by_count":2,"citation_normalized_percentile":{"value":0.59752409,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"2006","issue":"1","first_page":null,"last_page":null},"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":0.9876000285148621,"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/T12709","display_name":"Optimization and Mathematical Programming","score":0.9699000120162964,"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":0.8608660697937012},{"id":"https://openalex.org/keywords/duality","display_name":"Duality (order theory)","score":0.763340950012207},{"id":"https://openalex.org/keywords/convexity","display_name":"Convexity","score":0.7338534593582153},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.6338262557983398},{"id":"https://openalex.org/keywords/constraint","display_name":"Constraint (computer-aided design)","score":0.6047951579093933},{"id":"https://openalex.org/keywords/strong-duality","display_name":"Strong duality","score":0.6016845107078552},{"id":"https://openalex.org/keywords/dual","display_name":"Dual (grammatical number)","score":0.5131693482398987},{"id":"https://openalex.org/keywords/optimization-problem","display_name":"Optimization problem","score":0.4387015998363495},{"id":"https://openalex.org/keywords/directional-derivative","display_name":"Directional derivative","score":0.4171859323978424},{"id":"https://openalex.org/keywords/convex-function","display_name":"Convex function","score":0.41182148456573486},{"id":"https://openalex.org/keywords/regular-polygon","display_name":"Regular polygon","score":0.3949233591556549},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.13201552629470825},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.08867484331130981}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8608660697937012},{"id":"https://openalex.org/C2778023678","wikidata":"https://www.wikidata.org/wiki/Q554403","display_name":"Duality (order theory)","level":2,"score":0.763340950012207},{"id":"https://openalex.org/C72134830","wikidata":"https://www.wikidata.org/wiki/Q5166524","display_name":"Convexity","level":2,"score":0.7338534593582153},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.6338262557983398},{"id":"https://openalex.org/C2776036281","wikidata":"https://www.wikidata.org/wiki/Q48769818","display_name":"Constraint (computer-aided design)","level":2,"score":0.6047951579093933},{"id":"https://openalex.org/C163863214","wikidata":"https://www.wikidata.org/wiki/Q7624553","display_name":"Strong duality","level":3,"score":0.6016845107078552},{"id":"https://openalex.org/C2780980858","wikidata":"https://www.wikidata.org/wiki/Q110022","display_name":"Dual (grammatical number)","level":2,"score":0.5131693482398987},{"id":"https://openalex.org/C137836250","wikidata":"https://www.wikidata.org/wiki/Q984063","display_name":"Optimization problem","level":2,"score":0.4387015998363495},{"id":"https://openalex.org/C2011187","wikidata":"https://www.wikidata.org/wiki/Q383851","display_name":"Directional derivative","level":2,"score":0.4171859323978424},{"id":"https://openalex.org/C145446738","wikidata":"https://www.wikidata.org/wiki/Q319913","display_name":"Convex function","level":3,"score":0.41182148456573486},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.3949233591556549},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.13201552629470825},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.08867484331130981},{"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/C124952713","wikidata":"https://www.wikidata.org/wiki/Q8242","display_name":"Literature","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/C142362112","wikidata":"https://www.wikidata.org/wiki/Q735","display_name":"Art","level":0,"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":5,"locations":[{"id":"doi:10.1155/ijmms/2006/30459","is_oa":true,"landing_page_url":"https://doi.org/10.1155/ijmms/2006/30459","pdf_url":"https://downloads.hindawi.com/journals/ijmms/2006/030459.pdf","source":{"id":"https://openalex.org/S151851439","display_name":"International Journal of Mathematics and Mathematical Sciences","issn_l":"0161-1712","issn":["0161-1712","1687-0425"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310319869","host_organization_name":"Hindawi Publishing Corporation","host_organization_lineage":["https://openalex.org/P4310319869","https://openalex.org/P4310320595"],"host_organization_lineage_names":["Hindawi Publishing Corporation","Wiley"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Mathematics and Mathematical Sciences","raw_type":"journal-article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.503.8735","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.503.8735","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.math.ethz.ch/EMIS/journals/HOA/IJMMS/Volume2006/30459.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.621.4507","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.621.4507","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.emis.de/journals/HOA/IJMMS/Volume2006/030459.pdf","raw_type":"text"},{"id":"pmh:oai:RePEc:hin:jijmms:030459","is_oa":false,"landing_page_url":"http://downloads.hindawi.com/journals/IJMMS/2006/030459.xml","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:doaj.org/article:d800728254184cccbccfaf78442fe03a","is_oa":false,"landing_page_url":"https://doaj.org/article/d800728254184cccbccfaf78442fe03a","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":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"International Journal of Mathematics and Mathematical Sciences, Vol 2006 (2006)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.1155/ijmms/2006/30459","is_oa":true,"landing_page_url":"https://doi.org/10.1155/ijmms/2006/30459","pdf_url":"https://downloads.hindawi.com/journals/ijmms/2006/030459.pdf","source":{"id":"https://openalex.org/S151851439","display_name":"International Journal of Mathematics and Mathematical Sciences","issn_l":"0161-1712","issn":["0161-1712","1687-0425"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310319869","host_organization_name":"Hindawi Publishing Corporation","host_organization_lineage":["https://openalex.org/P4310319869","https://openalex.org/P4310320595"],"host_organization_lineage_names":["Hindawi Publishing Corporation","Wiley"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Mathematics and Mathematical Sciences","raw_type":"journal-article"},"sustainable_development_goals":[{"id":"https://metadata.un.org/sdg/10","display_name":"Reduced inequalities","score":0.5699999928474426}],"awards":[{"id":"https://openalex.org/G151909099","display_name":null,"funder_award_id":"821134","funder_id":"https://openalex.org/F4320324152","funder_display_name":"University of Isfahan"}],"funders":[{"id":"https://openalex.org/F4320324152","display_name":"University of Isfahan","ror":"https://ror.org/05h9t7759"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W1977158646.pdf","grobid_xml":"https://content.openalex.org/works/W1977158646.grobid-xml"},"referenced_works_count":10,"referenced_works":["https://openalex.org/W1551360398","https://openalex.org/W1561610472","https://openalex.org/W1960239420","https://openalex.org/W2010215636","https://openalex.org/W2020065878","https://openalex.org/W2111948593","https://openalex.org/W2561178549","https://openalex.org/W2787523326","https://openalex.org/W3000651840","https://openalex.org/W3039948875"],"related_works":["https://openalex.org/W2392181631","https://openalex.org/W3213165575","https://openalex.org/W2406638882","https://openalex.org/W2391797119","https://openalex.org/W4307261727","https://openalex.org/W2136200239","https://openalex.org/W2080941243","https://openalex.org/W2362763982","https://openalex.org/W4283709143","https://openalex.org/W4224273196"],"abstract_inverted_index":{"We":[0],"are":[1,46,70],"concerned":[2],"with":[3,9],"a":[4,52,73],"nonsmooth":[5],"multiobjective":[6],"optimization":[7],"problem":[8,60],"inequality":[10],"constraints.":[11],"In":[12],"order":[13],"to":[14,62],"obtain":[15],"our":[16],"main":[17],"results,":[18],"we":[19,56],"give":[20],"the":[21,24,30,35,49,58,63],"definitions":[22],"of":[23,51],"generalized":[25,31,37],"convex":[26],"functions":[27],"based":[28],"on":[29],"directional":[32],"derivative.":[33],"Under":[34],"above":[36],"convexity":[38],"assumptions,":[39],"sufficient":[40],"and":[41,66],"necessary":[42],"conditions":[43],"for":[44],"optimality":[45],"given":[47],"without":[48,72],"need":[50],"constraint":[53,74],"qualification.":[54,75],"Then":[55],"formulate":[57],"dual":[59],"corresponding":[61],"primal":[64],"problem,":[65],"some":[67],"duality":[68],"results":[69],"obtained":[71]},"counts_by_year":[],"updated_date":"2026-08-15T07:11:24.734988","created_date":"2025-10-10T00:00:00"}
