{"id":"https://openalex.org/W2006148793","doi":"https://doi.org/10.1080/1055678041001697659","title":"Lower-order penalty methods for mathematical programs with complementarity constraints","display_name":"Lower-order penalty methods for mathematical programs with complementarity constraints","publication_year":2004,"publication_date":"2004-11-17","ids":{"openalex":"https://openalex.org/W2006148793","doi":"https://doi.org/10.1080/1055678041001697659","mag":"2006148793"},"language":"en","primary_location":{"id":"doi:10.1080/1055678041001697659","is_oa":false,"landing_page_url":"https://doi.org/10.1080/1055678041001697659","pdf_url":null,"source":{"id":"https://openalex.org/S103047102","display_name":"Optimization methods & software","issn_l":"1026-7670","issn":["1026-7670","1029-4937","1055-6788"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320547","host_organization_name":"Taylor & Francis","host_organization_lineage":["https://openalex.org/P4310320547"],"host_organization_lineage_names":["Taylor & Francis"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Optimization Methods and Software","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5024523994","display_name":"Xiaoqi Yang","orcid":"https://orcid.org/0000-0002-5583-4032"},"institutions":[{"id":"https://openalex.org/I14243506","display_name":"Hong Kong Polytechnic University","ror":"https://ror.org/0030zas98","country_code":"HK","type":"education","lineage":["https://openalex.org/I14243506"]}],"countries":["HK"],"is_corresponding":true,"raw_author_name":"X. Q. Yang","raw_affiliation_strings":["Department of Applied Mathematics , The Hong Kong Polytechnic University , Kowloon, Hong Kong, China","[Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong, China]"],"affiliations":[{"raw_affiliation_string":"Department of Applied Mathematics , The Hong Kong Polytechnic University , Kowloon, Hong Kong, China","institution_ids":["https://openalex.org/I14243506"]},{"raw_affiliation_string":"[Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong, China]","institution_ids":["https://openalex.org/I14243506"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5028578315","display_name":"Xiaoxu Huang","orcid":"https://orcid.org/0000-0002-4635-6112"},"institutions":[{"id":"https://openalex.org/I14243506","display_name":"Hong Kong Polytechnic University","ror":"https://ror.org/0030zas98","country_code":"HK","type":"education","lineage":["https://openalex.org/I14243506"]}],"countries":["HK"],"is_corresponding":false,"raw_author_name":"X. X. Huang","raw_affiliation_strings":["Department of Applied Mathematics , The Hong Kong Polytechnic University , Kowloon, Hong Kong, China ; Department of Mathematics and Computer Science , Chongqing Normal University , Chongqing, 400047, China"],"affiliations":[{"raw_affiliation_string":"Department of Applied Mathematics , The Hong Kong Polytechnic University , Kowloon, Hong Kong, China ; Department of Mathematics and Computer Science , Chongqing Normal University , Chongqing, 400047, China","institution_ids":["https://openalex.org/I14243506"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5024523994"],"corresponding_institution_ids":["https://openalex.org/I14243506"],"apc_list":null,"apc_paid":null,"fwci":1.3487,"has_fulltext":false,"cited_by_count":22,"citation_normalized_percentile":{"value":0.79830836,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"19","issue":"6","first_page":"693","last_page":"720"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10545","display_name":"Optimization and Variational Analysis","score":0.9998999834060669,"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":0.9998999834060669,"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":0.9983000159263611,"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":0.9837999939918518,"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/penalty-method","display_name":"Penalty method","score":0.8081945180892944},{"id":"https://openalex.org/keywords/limit-point","display_name":"Limit point","score":0.7679409980773926},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7140140533447266},{"id":"https://openalex.org/keywords/complementarity-theory","display_name":"Complementarity theory","score":0.6638056039810181},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.6462677121162415},{"id":"https://openalex.org/keywords/lipschitz-continuity","display_name":"Lipschitz continuity","score":0.6253918409347534},{"id":"https://openalex.org/keywords/mixed-complementarity-problem","display_name":"Mixed complementarity problem","score":0.6065578460693359},{"id":"https://openalex.org/keywords/limit","display_name":"Limit (mathematics)","score":0.582131028175354},{"id":"https://openalex.org/keywords/stationary-point","display_name":"Stationary point","score":0.5580883026123047},{"id":"https://openalex.org/keywords/complementarity","display_name":"Complementarity (molecular biology)","score":0.4778408706188202},{"id":"https://openalex.org/keywords/point","display_name":"Point (geometry)","score":0.4596080780029297},{"id":"https://openalex.org/keywords/optimization-problem","display_name":"Optimization problem","score":0.43392547965049744},{"id":"https://openalex.org/keywords/constraint","display_name":"Constraint (computer-aided design)","score":0.42906302213668823},{"id":"https://openalex.org/keywords/order","display_name":"Order (exchange)","score":0.41118869185447693},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.3743932545185089},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.164808452129364},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.11967280507087708}],"concepts":[{"id":"https://openalex.org/C6180225","wikidata":"https://www.wikidata.org/wiki/Q3411771","display_name":"Penalty method","level":2,"score":0.8081945180892944},{"id":"https://openalex.org/C63548660","wikidata":"https://www.wikidata.org/wiki/Q858223","display_name":"Limit point","level":2,"score":0.7679409980773926},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7140140533447266},{"id":"https://openalex.org/C85404239","wikidata":"https://www.wikidata.org/wiki/Q5156417","display_name":"Complementarity theory","level":3,"score":0.6638056039810181},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.6462677121162415},{"id":"https://openalex.org/C22324862","wikidata":"https://www.wikidata.org/wiki/Q652707","display_name":"Lipschitz continuity","level":2,"score":0.6253918409347534},{"id":"https://openalex.org/C77196050","wikidata":"https://www.wikidata.org/wiki/Q6883962","display_name":"Mixed complementarity problem","level":3,"score":0.6065578460693359},{"id":"https://openalex.org/C151201525","wikidata":"https://www.wikidata.org/wiki/Q177239","display_name":"Limit (mathematics)","level":2,"score":0.582131028175354},{"id":"https://openalex.org/C189237950","wikidata":"https://www.wikidata.org/wiki/Q2500758","display_name":"Stationary point","level":2,"score":0.5580883026123047},{"id":"https://openalex.org/C202269582","wikidata":"https://www.wikidata.org/wiki/Q2644277","display_name":"Complementarity (molecular biology)","level":2,"score":0.4778408706188202},{"id":"https://openalex.org/C28719098","wikidata":"https://www.wikidata.org/wiki/Q44946","display_name":"Point (geometry)","level":2,"score":0.4596080780029297},{"id":"https://openalex.org/C137836250","wikidata":"https://www.wikidata.org/wiki/Q984063","display_name":"Optimization problem","level":2,"score":0.43392547965049744},{"id":"https://openalex.org/C2776036281","wikidata":"https://www.wikidata.org/wiki/Q48769818","display_name":"Constraint (computer-aided design)","level":2,"score":0.42906302213668823},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.41118869185447693},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3743932545185089},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.164808452129364},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.11967280507087708},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"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/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C54355233","wikidata":"https://www.wikidata.org/wiki/Q7162","display_name":"Genetics","level":1,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","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/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1080/1055678041001697659","is_oa":false,"landing_page_url":"https://doi.org/10.1080/1055678041001697659","pdf_url":null,"source":{"id":"https://openalex.org/S103047102","display_name":"Optimization methods & software","issn_l":"1026-7670","issn":["1026-7670","1029-4937","1055-6788"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320547","host_organization_name":"Taylor & Francis","host_organization_lineage":["https://openalex.org/P4310320547"],"host_organization_lineage_names":["Taylor & Francis"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Optimization Methods and Software","raw_type":"journal-article"},{"id":"pmh:oai:ira.lib.polyu.edu.hk:10397/15079","is_oa":false,"landing_page_url":"http://hdl.handle.net/10397/15079","pdf_url":null,"source":{"id":"https://openalex.org/S4306400205","display_name":"PolyU Institutional Research Archive (Hong Kong Polytechnic University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I14243506","host_organization_name":"Hong Kong Polytechnic University","host_organization_lineage":["https://openalex.org/I14243506"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Journal/Magazine Article"}],"best_oa_location":null,"sustainable_development_goals":[{"display_name":"Peace, Justice and strong institutions","score":0.7900000214576721,"id":"https://metadata.un.org/sdg/16"}],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":15,"referenced_works":["https://openalex.org/W217831350","https://openalex.org/W1966095601","https://openalex.org/W1972577527","https://openalex.org/W1976886395","https://openalex.org/W2005604902","https://openalex.org/W2008648847","https://openalex.org/W2019727142","https://openalex.org/W2037678919","https://openalex.org/W2042838878","https://openalex.org/W2056723044","https://openalex.org/W2077136942","https://openalex.org/W2082451773","https://openalex.org/W2133374626","https://openalex.org/W4246495354","https://openalex.org/W4285719527"],"related_works":["https://openalex.org/W2014001084","https://openalex.org/W2361513852","https://openalex.org/W2391163278","https://openalex.org/W1970914972","https://openalex.org/W2122617831","https://openalex.org/W2014704055","https://openalex.org/W2050897335","https://openalex.org/W2006148793","https://openalex.org/W2909617503","https://openalex.org/W2095246905"],"abstract_inverted_index":{"In":[0],"this":[1],"article,":[2],"a":[3,14,76,88,103,108,124,149],"smooth":[4,77],"mathematical":[5],"program":[6],"with":[7],"complementarity":[8,136],"constraints":[9],"(MPCC)":[10],"is":[11,28,102,119],"reformulated":[12],"as":[13],"non-smooth":[15],"constrained":[16,34],"optimization":[17,35,39],"problem":[18,36],"by":[19],"using":[20,75],"the":[21,32,61,72,84,94,112,116,140,144,163,166],"Fischer-Burmeister":[22],"function.":[23],"A":[24],"lower-order":[25],"penalty":[26,42,65,73,100],"method":[27],"applied":[29],"to":[30,54,121,159],"transform":[31],"resulted":[33],"into":[37],"unconstrained":[38],"problems.":[40],"Lower-order":[41],"functions":[43],"may":[44],"not":[45],"be":[46],"locally":[47],"Lipschitz.":[48],"However,":[49],"they":[50],"require":[51],"weaker":[52],"conditions":[53,70,98],"guarantee":[55],"an":[56,132],"exact":[57],"penalization":[58],"property":[59],"than":[60],"classical":[62],"l":[63],"1":[64],"functions.":[66],"We":[67],"derive":[68],"optimality":[69,97],"for":[71,129],"problems":[74,101],"approximate":[78],"variational":[79],"principle,":[80],"and":[81,123,131,161],"establish":[82],"that":[83,92],"limit":[85,117,141,145],"point":[86,106,118,146],"of":[87,90,99,111,153,165],"sequence":[89],"points":[91],"satisfy":[93],"second-order":[95,150],"necessary":[96,151],"strongly":[104],"stationary":[105],"(hence":[107],"B-stationary":[109],"point)":[110],"original":[113],"MPCC":[114,130],"if":[115],"feasible":[120],"MPCC,":[122],"linear":[125],"independence":[126],"constraint":[127],"qualification":[128],"upper":[133],"level":[134],"strict":[135],"condition":[137,152],"hold":[138],"at":[139],"point.":[142],"Furthermore,":[143],"also":[147],"satisfies":[148],"MPCC.":[154],"Numerical":[155],"examples":[156],"are":[157],"presented":[158],"demonstrate":[160],"compare":[162],"effectiveness":[164],"proposed":[167],"methods.":[168]},"counts_by_year":[{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":2},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":1},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":1},{"year":2015,"cited_by_count":2},{"year":2014,"cited_by_count":2},{"year":2012,"cited_by_count":2}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}