{"id":"https://openalex.org/W3204640158","doi":"https://doi.org/10.1007/s00186-022-00798-6","title":"Bilevel hyperparameter optimization for support vector classification: theoretical analysis and a solution method","display_name":"Bilevel hyperparameter optimization for support vector classification: theoretical analysis and a solution method","publication_year":2022,"publication_date":"2022-08-26","ids":{"openalex":"https://openalex.org/W3204640158","doi":"https://doi.org/10.1007/s00186-022-00798-6","mag":"3204640158"},"language":"en","primary_location":{"id":"doi:10.1007/s00186-022-00798-6","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00186-022-00798-6","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00186-022-00798-6.pdf","source":{"id":"https://openalex.org/S126160977","display_name":"Mathematical Methods of Operations Research","issn_l":"1432-2994","issn":["1432-2994","1432-5217"],"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":"Mathematical Methods of Operations Research","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/s00186-022-00798-6.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5077587421","display_name":"Qingna Li","orcid":"https://orcid.org/0000-0002-1397-6200"},"institutions":[{"id":"https://openalex.org/I125839683","display_name":"Beijing Institute of Technology","ror":"https://ror.org/01skt4w74","country_code":"CN","type":"education","lineage":["https://openalex.org/I125839683","https://openalex.org/I890469752"]},{"id":"https://openalex.org/I890469752","display_name":"Ministry of Industry and Information Technology","ror":"https://ror.org/0385nmy68","country_code":"CN","type":"government","lineage":["https://openalex.org/I890469752"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Qingna Li","raw_affiliation_strings":["School of Mathematics and Statistics/Beijing Key Laboratory on MCAACI/Key Laboratory of Mathematical Theory and Computation in Information Security, Beijing Institute of Technology, Beijing 100081, People's Republic of China","School of Mathematics and Statistics/Beijing Key Laboratory on MCAACI/Key Laboratory of Mathematical Theory and Computation in Information Security, Beijing Institute of Technology, Beijing, 100081, People's Republic of China"],"affiliations":[{"raw_affiliation_string":"School of Mathematics and Statistics/Beijing Key Laboratory on MCAACI/Key Laboratory of Mathematical Theory and Computation in Information Security, Beijing Institute of Technology, Beijing 100081, People's Republic of China","institution_ids":["https://openalex.org/I890469752"]},{"raw_affiliation_string":"School of Mathematics and Statistics/Beijing Key Laboratory on MCAACI/Key Laboratory of Mathematical Theory and Computation in Information Security, Beijing Institute of Technology, Beijing, 100081, People's Republic of China","institution_ids":["https://openalex.org/I125839683"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5100332655","display_name":"Zhen Li","orcid":"https://orcid.org/0000-0003-1166-9482"},"institutions":[{"id":"https://openalex.org/I125839683","display_name":"Beijing Institute of Technology","ror":"https://ror.org/01skt4w74","country_code":"CN","type":"education","lineage":["https://openalex.org/I125839683","https://openalex.org/I890469752"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Zhen Li","raw_affiliation_strings":["School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, People's Republic of China","School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, 100081, People's Republic of China"],"affiliations":[{"raw_affiliation_string":"School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, People's Republic of China","institution_ids":["https://openalex.org/I125839683"]},{"raw_affiliation_string":"School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, 100081, People's Republic of China","institution_ids":["https://openalex.org/I125839683"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5035136256","display_name":"Alain B. Zemkoho","orcid":"https://orcid.org/0000-0003-1265-4178"},"institutions":[{"id":"https://openalex.org/I43439940","display_name":"University of Southampton","ror":"https://ror.org/01ryk1543","country_code":"GB","type":"education","lineage":["https://openalex.org/I43439940"]}],"countries":["GB"],"is_corresponding":true,"raw_author_name":"Alain Zemkoho","raw_affiliation_strings":["School of Mathematical Sciences, University of Southampton, Southampton SO17 1BJ, UK 123","School of Mathematical Sciences, University of Southampton, Southampton, SO17 1BJ, UK"],"affiliations":[{"raw_affiliation_string":"School of Mathematical Sciences, University of Southampton, Southampton SO17 1BJ, UK 123","institution_ids":["https://openalex.org/I43439940"]},{"raw_affiliation_string":"School of Mathematical Sciences, University of Southampton, Southampton, SO17 1BJ, UK","institution_ids":["https://openalex.org/I43439940"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5035136256"],"corresponding_institution_ids":["https://openalex.org/I43439940"],"apc_list":{"value":2290,"currency":"EUR","value_usd":2890},"apc_paid":{"value":2290,"currency":"EUR","value_usd":2890},"fwci":1.9148,"has_fulltext":true,"cited_by_count":6,"citation_normalized_percentile":{"value":0.81600985,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":97},"biblio":{"volume":"96","issue":"3","first_page":"315","last_page":"350"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.9957000017166138,"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"}},"topics":[{"id":"https://openalex.org/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.9957000017166138,"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/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.9945999979972839,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"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/T12072","display_name":"Machine Learning and Algorithms","score":0.9864000082015991,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/hyperparameter","display_name":"Hyperparameter","score":0.8479748964309692},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.5848170518875122},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.5335484147071838},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.49811363220214844},{"id":"https://openalex.org/keywords/relaxation","display_name":"Relaxation (psychology)","score":0.45822250843048096},{"id":"https://openalex.org/keywords/bilevel-optimization","display_name":"Bilevel optimization","score":0.4572640657424927},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.43869784474372864},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.3743586242198944},{"id":"https://openalex.org/keywords/optimization-problem","display_name":"Optimization problem","score":0.335005521774292}],"concepts":[{"id":"https://openalex.org/C8642999","wikidata":"https://www.wikidata.org/wiki/Q4171168","display_name":"Hyperparameter","level":2,"score":0.8479748964309692},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.5848170518875122},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.5335484147071838},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.49811363220214844},{"id":"https://openalex.org/C2776029896","wikidata":"https://www.wikidata.org/wiki/Q3935810","display_name":"Relaxation (psychology)","level":2,"score":0.45822250843048096},{"id":"https://openalex.org/C3309286","wikidata":"https://www.wikidata.org/wiki/Q4907693","display_name":"Bilevel optimization","level":3,"score":0.4572640657424927},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.43869784474372864},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.3743586242198944},{"id":"https://openalex.org/C137836250","wikidata":"https://www.wikidata.org/wiki/Q984063","display_name":"Optimization problem","level":2,"score":0.335005521774292},{"id":"https://openalex.org/C15744967","wikidata":"https://www.wikidata.org/wiki/Q9418","display_name":"Psychology","level":0,"score":0.0},{"id":"https://openalex.org/C77805123","wikidata":"https://www.wikidata.org/wiki/Q161272","display_name":"Social psychology","level":1,"score":0.0},{"id":"https://openalex.org/C50522688","wikidata":"https://www.wikidata.org/wiki/Q189833","display_name":"Economic growth","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":3,"locations":[{"id":"doi:10.1007/s00186-022-00798-6","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00186-022-00798-6","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00186-022-00798-6.pdf","source":{"id":"https://openalex.org/S126160977","display_name":"Mathematical Methods of Operations Research","issn_l":"1432-2994","issn":["1432-2994","1432-5217"],"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":"Mathematical Methods of Operations Research","raw_type":"journal-article"},{"id":"pmh:oai:eprints.soton.ac.uk:473404","is_oa":true,"landing_page_url":"http://doi.org/10.1007/s00186-022-00798-6>).","pdf_url":"https://eprints.soton.ac.uk/473404/1/s00186_022_00798_6.pdf","source":{"id":"https://openalex.org/S4306401020","display_name":"ePrints Soton (University of Southampton)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I43439940","host_organization_name":"University of Southampton","host_organization_lineage":["https://openalex.org/I43439940"],"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":"PeerReviewed"},{"id":"pmh:oai:RePEc:spr:mathme:v:96:y:2022:i:3:d:10.1007_s00186-022-00798-6","is_oa":false,"landing_page_url":"http://link.springer.com/10.1007/s00186-022-00798-6","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/s00186-022-00798-6","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00186-022-00798-6","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00186-022-00798-6.pdf","source":{"id":"https://openalex.org/S126160977","display_name":"Mathematical Methods of Operations Research","issn_l":"1432-2994","issn":["1432-2994","1432-5217"],"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":"Mathematical Methods of Operations Research","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G1223493200","display_name":null,"funder_award_id":"EP/N510129/1.","funder_id":"https://openalex.org/F4320334627","funder_display_name":"Engineering and Physical Sciences Research Council"},{"id":"https://openalex.org/G1231421488","display_name":null,"funder_award_id":"under","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G1309768829","display_name":null,"funder_award_id":"EP/N510129/1.","funder_id":"https://openalex.org/F4320313467","funder_display_name":"Alan Turing Institute"},{"id":"https://openalex.org/G1633672591","display_name":null,"funder_award_id":"N510129","funder_id":"https://openalex.org/F4320334627","funder_display_name":"Engineering and Physical Sciences Research Council"},{"id":"https://openalex.org/G2087396116","display_name":null,"funder_award_id":"China","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G3219617319","display_name":"Approximation theory for two-level value functions with applications","funder_award_id":"EP/V049038/1","funder_id":"https://openalex.org/F4320334627","funder_display_name":"Engineering and Physical Sciences Research Council"},{"id":"https://openalex.org/G3317480652","display_name":null,"funder_award_id":"Science","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G3330119427","display_name":null,"funder_award_id":"EPSRC","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G3734132368","display_name":null,"funder_award_id":"12071032","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G3855435662","display_name":null,"funder_award_id":"N510129","funder_id":"https://openalex.org/F4320313467","funder_display_name":"Alan Turing Institute"},{"id":"https://openalex.org/G4072067573","display_name":null,"funder_award_id":"EP/N510129/1","funder_id":"https://openalex.org/F4320313467","funder_display_name":"Alan Turing Institute"},{"id":"https://openalex.org/G444630160","display_name":null,"funder_award_id":"EP/N510129/","funder_id":"https://openalex.org/F4320334627","funder_display_name":"Engineering and Physical Sciences Research Council"},{"id":"https://openalex.org/G4500485203","display_name":"The Alan Turing Institute","funder_award_id":"EP/N510129/1","funder_id":"https://openalex.org/F4320334627","funder_display_name":"Engineering and Physical Sciences Research Council"},{"id":"https://openalex.org/G6760021811","display_name":null,"funder_award_id":"EP/V049038/1","funder_id":"https://openalex.org/F4320334627","funder_display_name":"Engineering and Physical Sciences Research Council"},{"id":"https://openalex.org/G777199097","display_name":null,"funder_award_id":"EPSRC grant EP/N510129/1","funder_id":"https://openalex.org/F4320334627","funder_display_name":"Engineering and Physical Sciences Research Council"},{"id":"https://openalex.org/G8208342437","display_name":null,"funder_award_id":"1 and","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G8303412567","display_name":null,"funder_award_id":"EP/P022553/1","funder_id":"https://openalex.org/F4320334627","funder_display_name":"Engineering and Physical Sciences Research Council"},{"id":"https://openalex.org/G848032724","display_name":null,"funder_award_id":"Science","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G8493094889","display_name":null,"funder_award_id":"EP/V049038/1","funder_id":"https://openalex.org/F4320313467","funder_display_name":"Alan Turing Institute"},{"id":"https://openalex.org/G8597009950","display_name":null,"funder_award_id":"EP/N510129/1","funder_id":"https://openalex.org/F4320334627","funder_display_name":"Engineering and Physical Sciences Research Council"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320313467","display_name":"Alan Turing Institute","ror":"https://ror.org/035dkdb55"},{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320334627","display_name":"Engineering and Physical Sciences Research Council","ror":"https://ror.org/0439y7842"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3204640158.pdf","grobid_xml":"https://content.openalex.org/works/W3204640158.grobid-xml"},"referenced_works_count":65,"referenced_works":["https://openalex.org/W217831350","https://openalex.org/W1429889249","https://openalex.org/W1543439990","https://openalex.org/W1563088657","https://openalex.org/W1588486985","https://openalex.org/W1588628884","https://openalex.org/W1754531818","https://openalex.org/W1970163214","https://openalex.org/W1976886395","https://openalex.org/W1981276685","https://openalex.org/W1996676734","https://openalex.org/W2002511953","https://openalex.org/W2013910033","https://openalex.org/W2023446959","https://openalex.org/W2034039293","https://openalex.org/W2040164128","https://openalex.org/W2042838878","https://openalex.org/W2043610587","https://openalex.org/W2044241503","https://openalex.org/W2045826029","https://openalex.org/W2082988498","https://openalex.org/W2091825929","https://openalex.org/W2093663561","https://openalex.org/W2109346685","https://openalex.org/W2124022122","https://openalex.org/W2124659975","https://openalex.org/W2125993116","https://openalex.org/W2126813526","https://openalex.org/W2128255043","https://openalex.org/W2136627475","https://openalex.org/W2138218159","https://openalex.org/W2138849538","https://openalex.org/W2156417353","https://openalex.org/W2156909104","https://openalex.org/W2158001550","https://openalex.org/W2160712683","https://openalex.org/W2165966284","https://openalex.org/W2171126500","https://openalex.org/W2198630576","https://openalex.org/W2247017865","https://openalex.org/W2254218369","https://openalex.org/W2278603371","https://openalex.org/W2296242593","https://openalex.org/W2607956614","https://openalex.org/W2783380896","https://openalex.org/W2962818638","https://openalex.org/W2968575211","https://openalex.org/W2995990939","https://openalex.org/W3009517674","https://openalex.org/W3011655242","https://openalex.org/W3118521211","https://openalex.org/W3120771037","https://openalex.org/W3156469768","https://openalex.org/W3165240635","https://openalex.org/W3174488350","https://openalex.org/W3206178717","https://openalex.org/W3217444541","https://openalex.org/W4230674625","https://openalex.org/W4231147659","https://openalex.org/W4240373840","https://openalex.org/W4285719527","https://openalex.org/W4299564107","https://openalex.org/W4301791716","https://openalex.org/W6601926947","https://openalex.org/W6726548716"],"related_works":["https://openalex.org/W4297800546","https://openalex.org/W2804529069","https://openalex.org/W2963306862","https://openalex.org/W2808123587","https://openalex.org/W2890257643","https://openalex.org/W1495503294","https://openalex.org/W2161149697","https://openalex.org/W255338418","https://openalex.org/W2340378315","https://openalex.org/W2013038594"],"abstract_inverted_index":{"Abstract":[0],"Support":[1],"vector":[2,48],"classification":[3,13,22,49],"(SVC)":[4],"is":[5,30,104,137,159],"a":[6,51,108,121,142,160],"classical":[7],"and":[8,28,73],"well-performed":[9],"learning":[10],"method":[11,134],"for":[12,46,93],"problems.":[14],"A":[15],"regularization":[16],"parameter,":[17],"which":[18,56,158],"significantly":[19],"affects":[20],"the":[21,34,42,57,61,70,74,78,129,149,153,165,168,183,186,204],"performance,":[23],"has":[24],"to":[25,107,139,141,163],"be":[26],"chosen":[27],"this":[29,38,117,177,209],"usually":[31],"done":[32],"by":[33],"cross-validation":[35,71,124],"procedure.":[36],"In":[37,189],"paper,":[39],"we":[40,119,146],"reformulate":[41],"hyperparameter":[43],"selection":[44],"problem":[45,54,59],"support":[47],"as":[50],"bilevel":[52,101],"optimization":[53,102],"in":[55,96,208],"upper-level":[58],"minimizes":[60],"average":[62],"number":[63],"of":[64,152,167,176,185],"misclassified":[65],"data":[66,205],"points":[67],"over":[68,201],"all":[69,203],"folds,":[72],"lower-level":[75],"problems":[76],"are":[77],"$$l_1$$":[79],"<mml:math":[80],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">":[81],"<mml:msub>":[82],"<mml:mi>l</mml:mi>":[83],"<mml:mn>1</mml:mn>":[84],"</mml:msub>":[85],"</mml:math>":[86],"-loss":[87],"SVC":[88],"problems,":[89],"with":[90,111,192],"each":[91,94,173],"one":[92],"fold":[95],"T-fold":[97],"cross-validation.":[98],"The":[99],"resulting":[100],"model":[103],"then":[105],"converted":[106],"mathematical":[109],"program":[110],"equilibrium":[112],"constraints":[113],"(MPEC).":[114],"To":[115],"solve":[116],"MPEC,":[118],"propose":[120],"global":[122,132],"relaxation":[123,133],"algorithm":[125,196],"(GR\u2013CV)":[126],"based":[127],"on":[128],"well-know":[130],"Sholtes-type":[131],"(GRM).":[135],"It":[136],"proven":[138],"converge":[140],"C-stationary":[143],"point.":[144],"Moreover,":[145],"prove":[147],"that":[148],"MPEC-tailored":[150],"version":[151],"Mangasarian\u2013Fromovitz":[154],"constraint":[155],"qualification":[156],"(MFCQ),":[157],"key":[161],"property":[162],"guarantee":[164],"convergence":[166],"GRM,":[169],"automatically":[170],"holds":[171],"at":[172],"feasible":[174],"point":[175],"MPEC.":[178],"Extensive":[179],"numerical":[180],"results":[181],"verify":[182],"efficiency":[184],"proposed":[187],"approach.":[188],"particular,":[190],"compared":[191],"other":[193],"methods,":[194],"our":[195],"enjoys":[197],"superior":[198],"generalization":[199],"performance":[200],"almost":[202],"sets":[206],"used":[207],"paper.":[210]},"counts_by_year":[{"year":2024,"cited_by_count":3},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":2}],"updated_date":"2026-03-20T23:20:44.827607","created_date":"2025-10-10T00:00:00"}
