{"id":"https://openalex.org/W1982405519","doi":"https://doi.org/10.1080/10556780903178048","title":"Approximate optimality conditions and stopping criteria in canonical DC programming","display_name":"Approximate optimality conditions and stopping criteria in canonical DC programming","publication_year":2009,"publication_date":"2009-08-28","ids":{"openalex":"https://openalex.org/W1982405519","doi":"https://doi.org/10.1080/10556780903178048","mag":"1982405519"},"language":"en","primary_location":{"id":"doi:10.1080/10556780903178048","is_oa":false,"landing_page_url":"https://doi.org/10.1080/10556780903178048","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":true,"oa_status":"green","oa_url":"http://www.tandfonline.com/doi/abs/10.1080/10556780903178048","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5007636174","display_name":"Giancarlo Bigi","orcid":"https://orcid.org/0000-0002-0964-3294"},"institutions":[{"id":"https://openalex.org/I108290504","display_name":"University of Pisa","ror":"https://ror.org/03ad39j10","country_code":"IT","type":"education","lineage":["https://openalex.org/I108290504"]}],"countries":["IT"],"is_corresponding":false,"raw_author_name":"Giancarlo Bigi","raw_affiliation_strings":["Dipartimento di Informatica , Universit\u00e0 di Pisa , Largo B.Pontecorvo, 3, 56127, Pisa, Italia","Dipartimento di Informatica, Universit\u00e1 di Pisa, Pisa, Italia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Dipartimento di Informatica , Universit\u00e0 di Pisa , Largo B.Pontecorvo, 3, 56127, Pisa, Italia","institution_ids":["https://openalex.org/I108290504"]},{"raw_affiliation_string":"Dipartimento di Informatica, Universit\u00e1 di Pisa, Pisa, Italia","institution_ids":["https://openalex.org/I108290504"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5061303343","display_name":"Antonio Frangioni","orcid":"https://orcid.org/0000-0002-5704-3170"},"institutions":[{"id":"https://openalex.org/I108290504","display_name":"University of Pisa","ror":"https://ror.org/03ad39j10","country_code":"IT","type":"education","lineage":["https://openalex.org/I108290504"]}],"countries":["IT"],"is_corresponding":true,"raw_author_name":"Antonio Frangioni","raw_affiliation_strings":["Dipartimento di Informatica , Universit\u00e0 di Pisa , Largo B.Pontecorvo, 3, 56127, Pisa, Italia","Dipartimento di Informatica, Universit\u00e1 di Pisa, Pisa, Italia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Dipartimento di Informatica , Universit\u00e0 di Pisa , Largo B.Pontecorvo, 3, 56127, Pisa, Italia","institution_ids":["https://openalex.org/I108290504"]},{"raw_affiliation_string":"Dipartimento di Informatica, Universit\u00e1 di Pisa, Pisa, Italia","institution_ids":["https://openalex.org/I108290504"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5055297909","display_name":"Qinghua Zhang","orcid":"https://orcid.org/0000-0002-6154-4656"},"institutions":[{"id":"https://openalex.org/I108290504","display_name":"University of Pisa","ror":"https://ror.org/03ad39j10","country_code":"IT","type":"education","lineage":["https://openalex.org/I108290504"]}],"countries":["IT"],"is_corresponding":false,"raw_author_name":"Qinghua Zhang","raw_affiliation_strings":["Dipartimento di Informatica , Universit\u00e0 di Pisa , Largo B.Pontecorvo, 3, 56127, Pisa, Italia","Dipartimento di Informatica, Universit\u00e1 di Pisa, Pisa, Italia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Dipartimento di Informatica , Universit\u00e0 di Pisa , Largo B.Pontecorvo, 3, 56127, Pisa, Italia","institution_ids":["https://openalex.org/I108290504"]},{"raw_affiliation_string":"Dipartimento di Informatica, Universit\u00e1 di Pisa, Pisa, Italia","institution_ids":["https://openalex.org/I108290504"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5061303343"],"corresponding_institution_ids":["https://openalex.org/I108290504"],"apc_list":null,"apc_paid":null,"fwci":1.6398,"has_fulltext":false,"cited_by_count":5,"citation_normalized_percentile":{"value":0.8430904,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":96},"biblio":{"volume":"25","issue":"1","first_page":"19","last_page":"27"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11413","display_name":"Risk and Portfolio Optimization","score":0.9977999925613403,"subfield":{"id":"https://openalex.org/subfields/1803","display_name":"Management Science and Operations Research"},"field":{"id":"https://openalex.org/fields/18","display_name":"Decision Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},"topics":[{"id":"https://openalex.org/T11413","display_name":"Risk and Portfolio Optimization","score":0.9977999925613403,"subfield":{"id":"https://openalex.org/subfields/1803","display_name":"Management Science and Operations Research"},"field":{"id":"https://openalex.org/fields/18","display_name":"Decision Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},{"id":"https://openalex.org/T12288","display_name":"Optimization and Search Problems","score":0.9976999759674072,"subfield":{"id":"https://openalex.org/subfields/1705","display_name":"Computer Networks and Communications"},"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":0.98089998960495,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.6806777715682983},{"id":"https://openalex.org/keywords/optimal-stopping","display_name":"Optimal stopping","score":0.5969190001487732},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5934931039810181},{"id":"https://openalex.org/keywords/polyhedron","display_name":"Polyhedron","score":0.5656809210777283},{"id":"https://openalex.org/keywords/regular-polygon","display_name":"Regular polygon","score":0.5209763646125793},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.5129711031913757},{"id":"https://openalex.org/keywords/constraint","display_name":"Constraint (computer-aided design)","score":0.47987619042396545},{"id":"https://openalex.org/keywords/optimization-problem","display_name":"Optimization problem","score":0.47385695576667786},{"id":"https://openalex.org/keywords/task","display_name":"Task (project management)","score":0.45020806789398193},{"id":"https://openalex.org/keywords/value","display_name":"Value (mathematics)","score":0.42430180311203003},{"id":"https://openalex.org/keywords/convex-optimization","display_name":"Convex optimization","score":0.4186372756958008},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.38620492815971375},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.088213711977005}],"concepts":[{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.6806777715682983},{"id":"https://openalex.org/C99414536","wikidata":"https://www.wikidata.org/wiki/Q7098950","display_name":"Optimal stopping","level":2,"score":0.5969190001487732},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5934931039810181},{"id":"https://openalex.org/C54829058","wikidata":"https://www.wikidata.org/wiki/Q172937","display_name":"Polyhedron","level":2,"score":0.5656809210777283},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.5209763646125793},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.5129711031913757},{"id":"https://openalex.org/C2776036281","wikidata":"https://www.wikidata.org/wiki/Q48769818","display_name":"Constraint (computer-aided design)","level":2,"score":0.47987619042396545},{"id":"https://openalex.org/C137836250","wikidata":"https://www.wikidata.org/wiki/Q984063","display_name":"Optimization problem","level":2,"score":0.47385695576667786},{"id":"https://openalex.org/C2780451532","wikidata":"https://www.wikidata.org/wiki/Q759676","display_name":"Task (project management)","level":2,"score":0.45020806789398193},{"id":"https://openalex.org/C2776291640","wikidata":"https://www.wikidata.org/wiki/Q2912517","display_name":"Value (mathematics)","level":2,"score":0.42430180311203003},{"id":"https://openalex.org/C157972887","wikidata":"https://www.wikidata.org/wiki/Q463359","display_name":"Convex optimization","level":3,"score":0.4186372756958008},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.38620492815971375},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.088213711977005},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C187736073","wikidata":"https://www.wikidata.org/wiki/Q2920921","display_name":"Management","level":1,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","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/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1080/10556780903178048","is_oa":false,"landing_page_url":"https://doi.org/10.1080/10556780903178048","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:arpi.unipi.it:11568/136627","is_oa":true,"landing_page_url":"http://www.tandfonline.com/doi/abs/10.1080/10556780903178048","pdf_url":null,"source":{"id":"https://openalex.org/S4377196265","display_name":"CINECA IRIS Institutial research information system (University of Pisa)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I108290504","host_organization_name":"University of Pisa","host_organization_lineage":["https://openalex.org/I108290504"],"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":null,"raw_type":"info:eu-repo/semantics/article"}],"best_oa_location":{"id":"pmh:oai:arpi.unipi.it:11568/136627","is_oa":true,"landing_page_url":"http://www.tandfonline.com/doi/abs/10.1080/10556780903178048","pdf_url":null,"source":{"id":"https://openalex.org/S4377196265","display_name":"CINECA IRIS Institutial research information system (University of Pisa)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I108290504","host_organization_name":"University of Pisa","host_organization_lineage":["https://openalex.org/I108290504"],"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":null,"raw_type":"info:eu-repo/semantics/article"},"sustainable_development_goals":[{"score":0.4000000059604645,"display_name":"Partnerships for the goals","id":"https://metadata.un.org/sdg/17"}],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":18,"referenced_works":["https://openalex.org/W44939114","https://openalex.org/W46129499","https://openalex.org/W75157487","https://openalex.org/W1497465005","https://openalex.org/W1563945042","https://openalex.org/W1971867305","https://openalex.org/W2016408076","https://openalex.org/W2044800863","https://openalex.org/W2053427412","https://openalex.org/W2056371558","https://openalex.org/W2064451830","https://openalex.org/W2082476243","https://openalex.org/W2082639035","https://openalex.org/W2084182592","https://openalex.org/W2119993761","https://openalex.org/W2122565017","https://openalex.org/W3150183338","https://openalex.org/W4249840661"],"related_works":["https://openalex.org/W2971340537","https://openalex.org/W1584292662","https://openalex.org/W2046210743","https://openalex.org/W1857475410","https://openalex.org/W1511346678","https://openalex.org/W2030136392","https://openalex.org/W2797788337","https://openalex.org/W2738111169","https://openalex.org/W2951184467","https://openalex.org/W9822165"],"abstract_inverted_index":{"In":[0,31,96],"this":[1,68,97,163],"paper,":[2],"we":[3],"study":[4,103],"approximate":[5,93],"optimality":[6,34,57,94,123],"conditions":[7,35,124,160],"for":[8,21,28,36],"the":[9,29,56,81,104,107,110,114,117,120,139,151,155,165,175,187,190],"Canonical":[10],"DC":[11],"(CDC)":[12],"optimization":[13,47],"problem":[14,82],"and":[15,90,113],"their":[16],"relationships":[17,105],"with":[18,174,186],"stopping":[19,88,111,140],"criteria":[20,89,112,141],"a":[22,45,59,72,135,146,181,194],"large":[23,148],"class":[24],"of":[25,44,58,116,150,154,178,180],"solution":[26,62],"algorithms":[27],"problem.":[30,157],"fact,":[32],"global":[33],"CDC":[37,156,182],"are":[38],"very":[39],"often":[40],"restated":[41],"in":[42,70,109,130,138],"terms":[43],"non-convex":[46],"problem,":[48,183],"which":[49],"has":[50,63],"to":[51,64,78,85,92,102,132,145,170],"be":[52,65,171],"solved":[53],"each":[54],"time":[55],"given":[60],"tentative":[61],"checked.":[66],"Since":[67],"is":[69,100,164,193],"principle":[71],"costly":[73],"task,":[74],"it":[75,99],"makes":[76],"sense":[77],"only":[79],"solve":[80],"approximately,":[83],"leading":[84],"an":[86],"inexact":[87],"therefore":[91],"conditions.":[95],"framework,":[98],"important":[101],"between":[106],"approximation":[108,149],"quality":[115],"solutions":[118],"that":[119,134,162],"corresponding":[121],"approximated":[122],"may":[125],"eventually":[126],"accept":[127],"as":[128],"optimal,":[129],"order":[131],"ensure":[133],"small":[136],"tolerance":[137],"does":[142],"not":[143],"lead":[144],"disproportionally":[147],"optimal":[152],"value":[153],"We":[158],"develop":[159],"ensuring":[161],"case;":[166],"these":[167],"turn":[168],"out":[169],"closely":[172],"related":[173],"well-known":[176],"concept":[177],"regularity":[179],"actually":[184],"coinciding":[185],"latter":[188],"if":[189],"reverse-constraint":[191],"set":[192],"polyhedron.":[195]},"counts_by_year":[{"year":2019,"cited_by_count":1},{"year":2012,"cited_by_count":2}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}