{"id":"https://openalex.org/W4401122635","doi":"https://doi.org/10.1051/ro/2024157","title":"First and second order necessary optimality conditions for multiobjective programming with interval-valued objective functions on Riemannian manifolds","display_name":"First and second order necessary optimality conditions for multiobjective programming with interval-valued objective functions on Riemannian manifolds","publication_year":2024,"publication_date":"2024-07-30","ids":{"openalex":"https://openalex.org/W4401122635","doi":"https://doi.org/10.1051/ro/2024157"},"language":"en","primary_location":{"id":"doi:10.1051/ro/2024157","is_oa":true,"landing_page_url":"http://dx.doi.org/10.1051/ro/2024157","pdf_url":"https://www.rairo-ro.org/articles/ro/pdf/2024/05/ro240137.pdf","source":{"id":"https://openalex.org/S20631277","display_name":"RAIRO - Operations Research","issn_l":"0399-0559","issn":["0399-0559","1290-3868","2804-7303"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319748","host_organization_name":"EDP Sciences","host_organization_lineage":["https://openalex.org/P4310319748","https://openalex.org/P4310319982"],"host_organization_lineage_names":["EDP Sciences","Science Press"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"RAIRO - Operations Research","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://www.rairo-ro.org/articles/ro/pdf/2024/05/ro240137.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5103167704","display_name":"Hilal Ahmad Bhat","orcid":"https://orcid.org/0000-0002-8712-3905"},"institutions":[{"id":"https://openalex.org/I171210897","display_name":"Aligarh Muslim University","ror":"https://ror.org/03kw9gc02","country_code":"IN","type":"education","lineage":["https://openalex.org/I171210897"]}],"countries":["IN"],"is_corresponding":false,"raw_author_name":"Hilal Ahmad Bhat","raw_affiliation_strings":["Department of Mathematics, Aligarh Muslim University, Aligarh 202002, Uttar Pradesh, India"],"raw_orcid":"https://orcid.org/0000-0002-8712-3905","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Aligarh Muslim University, Aligarh 202002, Uttar Pradesh, India","institution_ids":["https://openalex.org/I171210897"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5009174556","display_name":"Akhlad Iqbal","orcid":"https://orcid.org/0000-0003-1932-2782"},"institutions":[{"id":"https://openalex.org/I171210897","display_name":"Aligarh Muslim University","ror":"https://ror.org/03kw9gc02","country_code":"IN","type":"education","lineage":["https://openalex.org/I171210897"]}],"countries":["IN"],"is_corresponding":true,"raw_author_name":"Akhlad Iqbal","raw_affiliation_strings":["Department of Mathematics, Aligarh Muslim University, Aligarh 202002, Uttar Pradesh, India"],"raw_orcid":"https://orcid.org/0000-0003-1932-2782","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Aligarh Muslim University, Aligarh 202002, Uttar Pradesh, India","institution_ids":["https://openalex.org/I171210897"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5102557731","display_name":"Mahwash Aftab","orcid":null},"institutions":[{"id":"https://openalex.org/I171210897","display_name":"Aligarh Muslim University","ror":"https://ror.org/03kw9gc02","country_code":"IN","type":"education","lineage":["https://openalex.org/I171210897"]}],"countries":["IN"],"is_corresponding":false,"raw_author_name":"Mahwash Aftab","raw_affiliation_strings":["Department of Mathematics, Aligarh Muslim University, Aligarh 202002, Uttar Pradesh, India"],"raw_orcid":"https://orcid.org/0009-0008-0561-9079","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Aligarh Muslim University, Aligarh 202002, Uttar Pradesh, India","institution_ids":["https://openalex.org/I171210897"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5009174556"],"corresponding_institution_ids":["https://openalex.org/I171210897"],"apc_list":{"value":0,"currency":"USD","value_usd":0},"apc_paid":null,"fwci":0.9654,"has_fulltext":true,"cited_by_count":4,"citation_normalized_percentile":{"value":0.77350852,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":94,"max":98},"biblio":{"volume":"58","issue":"5","first_page":"4259","last_page":"4276"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10545","display_name":"Optimization and Variational Analysis","score":0.9822999835014343,"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":0.9822999835014343,"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/T10969","display_name":"Water resources management and optimization","score":0.9603999853134155,"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"}},{"id":"https://openalex.org/T10928","display_name":"Probabilistic and Robust Engineering Design","score":0.9358000159263611,"subfield":{"id":"https://openalex.org/subfields/1804","display_name":"Statistics, Probability and Uncertainty"},"field":{"id":"https://openalex.org/fields/18","display_name":"Decision Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/differentiable-function","display_name":"Differentiable function","score":0.8434677124023438},{"id":"https://openalex.org/keywords/interval","display_name":"Interval (graph theory)","score":0.7091397047042847},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.6731626987457275},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6575266122817993},{"id":"https://openalex.org/keywords/constraint","display_name":"Constraint (computer-aided design)","score":0.5631790161132812},{"id":"https://openalex.org/keywords/order","display_name":"Order (exchange)","score":0.5376729965209961},{"id":"https://openalex.org/keywords/optimization-problem","display_name":"Optimization problem","score":0.48357102274894714},{"id":"https://openalex.org/keywords/range","display_name":"Range (aeronautics)","score":0.4371680021286011},{"id":"https://openalex.org/keywords/manifold","display_name":"Manifold (fluid mechanics)","score":0.43121108412742615},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.33666300773620605},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.1469556987285614},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.06317812204360962}],"concepts":[{"id":"https://openalex.org/C202615002","wikidata":"https://www.wikidata.org/wiki/Q783507","display_name":"Differentiable function","level":2,"score":0.8434677124023438},{"id":"https://openalex.org/C2778067643","wikidata":"https://www.wikidata.org/wiki/Q166507","display_name":"Interval (graph theory)","level":2,"score":0.7091397047042847},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.6731626987457275},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6575266122817993},{"id":"https://openalex.org/C2776036281","wikidata":"https://www.wikidata.org/wiki/Q48769818","display_name":"Constraint (computer-aided design)","level":2,"score":0.5631790161132812},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.5376729965209961},{"id":"https://openalex.org/C137836250","wikidata":"https://www.wikidata.org/wiki/Q984063","display_name":"Optimization problem","level":2,"score":0.48357102274894714},{"id":"https://openalex.org/C204323151","wikidata":"https://www.wikidata.org/wiki/Q905424","display_name":"Range (aeronautics)","level":2,"score":0.4371680021286011},{"id":"https://openalex.org/C529865628","wikidata":"https://www.wikidata.org/wiki/Q1790740","display_name":"Manifold (fluid mechanics)","level":2,"score":0.43121108412742615},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.33666300773620605},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.1469556987285614},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.06317812204360962},{"id":"https://openalex.org/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","level":1,"score":0.0},{"id":"https://openalex.org/C78519656","wikidata":"https://www.wikidata.org/wiki/Q101333","display_name":"Mechanical engineering","level":1,"score":0.0},{"id":"https://openalex.org/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"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/C10138342","wikidata":"https://www.wikidata.org/wiki/Q43015","display_name":"Finance","level":1,"score":0.0},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","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":2,"locations":[{"id":"doi:10.1051/ro/2024157","is_oa":true,"landing_page_url":"http://dx.doi.org/10.1051/ro/2024157","pdf_url":"https://www.rairo-ro.org/articles/ro/pdf/2024/05/ro240137.pdf","source":{"id":"https://openalex.org/S20631277","display_name":"RAIRO - Operations Research","issn_l":"0399-0559","issn":["0399-0559","1290-3868","2804-7303"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319748","host_organization_name":"EDP Sciences","host_organization_lineage":["https://openalex.org/P4310319748","https://openalex.org/P4310319982"],"host_organization_lineage_names":["EDP Sciences","Science Press"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"RAIRO - Operations Research","raw_type":"journal-article"},{"id":"pmh:oai:edpsciences.org:dkey/10.1051/ro/2024157","is_oa":true,"landing_page_url":"https://www.rairo-ro.org/10.1051/ro/2024157/pdf","pdf_url":"https://www.rairo-ro.org/articles/ro/pdf/forth/ro240137.pdf","source":{"id":"https://openalex.org/S4306400744","display_name":"Springer Link (Chiba Institute of Technology)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I8488066","host_organization_name":"Chiba Institute of Technology","host_organization_lineage":["https://openalex.org/I8488066"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"https://doi.org/10.1051/ro/2024157","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.1051/ro/2024157","is_oa":true,"landing_page_url":"http://dx.doi.org/10.1051/ro/2024157","pdf_url":"https://www.rairo-ro.org/articles/ro/pdf/2024/05/ro240137.pdf","source":{"id":"https://openalex.org/S20631277","display_name":"RAIRO - Operations Research","issn_l":"0399-0559","issn":["0399-0559","1290-3868","2804-7303"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319748","host_organization_name":"EDP Sciences","host_organization_lineage":["https://openalex.org/P4310319748","https://openalex.org/P4310319982"],"host_organization_lineage_names":["EDP Sciences","Science Press"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"RAIRO - Operations Research","raw_type":"journal-article"},"sustainable_development_goals":[{"score":0.7900000214576721,"display_name":"Peace, Justice and strong institutions","id":"https://metadata.un.org/sdg/16"}],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4401122635.pdf","grobid_xml":"https://content.openalex.org/works/W4401122635.grobid-xml"},"referenced_works_count":30,"referenced_works":["https://openalex.org/W23712394","https://openalex.org/W138935389","https://openalex.org/W1580809199","https://openalex.org/W1633883681","https://openalex.org/W1967779834","https://openalex.org/W1983057379","https://openalex.org/W1984358873","https://openalex.org/W2001359285","https://openalex.org/W2007556045","https://openalex.org/W2009054772","https://openalex.org/W2019604114","https://openalex.org/W2027449074","https://openalex.org/W2029840249","https://openalex.org/W2049830000","https://openalex.org/W2082584863","https://openalex.org/W2088735768","https://openalex.org/W2109289092","https://openalex.org/W2117463942","https://openalex.org/W2330816622","https://openalex.org/W2919444388","https://openalex.org/W2994979367","https://openalex.org/W3038961962","https://openalex.org/W3080237625","https://openalex.org/W4200283354","https://openalex.org/W4205293427","https://openalex.org/W4235476591","https://openalex.org/W4244641530","https://openalex.org/W4323539420","https://openalex.org/W4384825491","https://openalex.org/W4398135734"],"related_works":["https://openalex.org/W4285277090","https://openalex.org/W4327738859","https://openalex.org/W2348722996","https://openalex.org/W2334570605","https://openalex.org/W3181683615","https://openalex.org/W4286826125","https://openalex.org/W1633485514","https://openalex.org/W1604739066","https://openalex.org/W2115878407","https://openalex.org/W3135859832"],"abstract_inverted_index":{"The":[0,140],"growing":[1],"dependence":[2],"on":[3,90],"optimization":[4,43,50],"models":[5],"in":[6,52,68],"decision-making":[7],"has":[8],"created":[9],"a":[10,22,81,91],"demand":[11],"for":[12,48,80,111,133],"tools":[13],"that":[14,39,120],"can":[15],"facilitate":[16],"the":[17,63,100,121,165,168],"formulation":[18],"and":[19,28,74,107,113,125,130,135],"resolution":[20],"of":[21,25,65,155,167],"broader":[23],"range":[24],"real-world":[26],"processes":[27],"systems":[29],"associated":[30],"with":[31,55,85],"human":[32],"activity.":[33],"These":[34],"situations":[35],"often":[36],"involve":[37],"assumptions":[38],"diverge":[40],"from":[41],"traditional":[42],"methodologies.":[44],"One":[45],"viable":[46],"approach":[47],"addressing":[49],"problems":[51],"real-life":[53],"scenarios":[54],"uncertainty":[56],"is":[57],"interval-valued":[58,66,86],"optimization.":[59],"Taking":[60],"into":[61],"account":[62],"significance":[64],"optimization,":[67],"this":[69],"paper,":[70],"we":[71,98,118,159],"derive":[72],"first":[73,112,134,141],"second":[75,114,136,146],"order":[76,137,142,147],"necessary":[77,148],"optimality":[78],"conditions":[79,138,149],"multi-objective":[82],"programming":[83],"problem":[84],"objective":[87,101],"functions":[88,102],"defined":[89],"Riemannian":[92],"manifold.":[93],"To":[94],"establish":[95],"these":[96],"conditions,":[97],"consider":[99],"to":[103,163],"be":[104],"weakly":[105,109],"differentiable":[106,110,129,132],"twice":[108,131],"order,":[115],"respectively.":[116,139],"Additionally,":[117],"assume":[119],"constraints,":[122,127],"both":[123],"equality":[124],"inequality":[126],"are":[128,150],"as":[143,145],"well":[144],"derived":[151],"under":[152],"two":[153],"types":[154],"constraint":[156],"qualifications.":[157],"Furthermore,":[158],"provide":[160],"illustrative":[161],"examples":[162],"demonstrate":[164],"application":[166],"established":[169],"results.":[170]},"counts_by_year":[{"year":2026,"cited_by_count":2},{"year":2025,"cited_by_count":2}],"updated_date":"2026-08-28T12:50:07.497085","created_date":"2025-10-10T00:00:00"}
