{"id":"https://openalex.org/W2255400221","doi":"https://doi.org/10.1137/110856010","title":"A Complete Characterization of the Gap between Convexity and SOS-Convexity","display_name":"A Complete Characterization of the Gap between Convexity and SOS-Convexity","publication_year":2013,"publication_date":"2013-01-01","ids":{"openalex":"https://openalex.org/W2255400221","doi":"https://doi.org/10.1137/110856010","mag":"2255400221"},"language":"en","primary_location":{"id":"doi:10.1137/110856010","is_oa":false,"landing_page_url":"https://doi.org/10.1137/110856010","pdf_url":null,"source":{"id":"https://openalex.org/S928796702","display_name":"SIAM Journal on Optimization","issn_l":"1052-6234","issn":["1052-6234","1095-7189"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Optimization","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1111.4587","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":null,"display_name":"Amir Ali Ahmadi","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Amir Ali Ahmadi","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":null,"display_name":"Pablo A. Parrilo","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Pablo A. Parrilo","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":4.6998,"has_fulltext":false,"cited_by_count":61,"citation_normalized_percentile":{"value":0.95270678,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":100},"biblio":{"volume":"23","issue":"2","first_page":"811","last_page":"833"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.6209999918937683,"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":0.6209999918937683,"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/T11435","display_name":"Polynomial and algebraic computation","score":0.1817999929189682,"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/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.019700000062584877,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"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/convexity","display_name":"Convexity","score":0.7897999882698059},{"id":"https://openalex.org/keywords/characterization","display_name":"Characterization (materials science)","score":0.5927000045776367},{"id":"https://openalex.org/keywords/degree","display_name":"Degree (music)","score":0.5536999702453613},{"id":"https://openalex.org/keywords/regular-polygon","display_name":"Regular polygon","score":0.47110000252723694},{"id":"https://openalex.org/keywords/algebraic-number","display_name":"Algebraic number","score":0.4634999930858612},{"id":"https://openalex.org/keywords/convex-set","display_name":"Convex set","score":0.45419999957084656},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.4251999855041504},{"id":"https://openalex.org/keywords/subderivative","display_name":"Subderivative","score":0.39489999413490295}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.9014999866485596},{"id":"https://openalex.org/C72134830","wikidata":"https://www.wikidata.org/wiki/Q5166524","display_name":"Convexity","level":2,"score":0.7897999882698059},{"id":"https://openalex.org/C2780841128","wikidata":"https://www.wikidata.org/wiki/Q5073781","display_name":"Characterization (materials science)","level":2,"score":0.5927000045776367},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.5536999702453613},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5339000225067139},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.47110000252723694},{"id":"https://openalex.org/C9376300","wikidata":"https://www.wikidata.org/wiki/Q168817","display_name":"Algebraic number","level":2,"score":0.4634999930858612},{"id":"https://openalex.org/C49870271","wikidata":"https://www.wikidata.org/wiki/Q193657","display_name":"Convex set","level":4,"score":0.45419999957084656},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.4251999855041504},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4108000099658966},{"id":"https://openalex.org/C200661725","wikidata":"https://www.wikidata.org/wiki/Q289500","display_name":"Subderivative","level":4,"score":0.39489999413490295},{"id":"https://openalex.org/C12108790","wikidata":"https://www.wikidata.org/wiki/Q2234833","display_name":"Convex analysis","level":4,"score":0.38370001316070557},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.36730000376701355},{"id":"https://openalex.org/C101901036","wikidata":"https://www.wikidata.org/wiki/Q2269096","display_name":"Semidefinite programming","level":2,"score":0.3264999985694885},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3172999918460846},{"id":"https://openalex.org/C145446738","wikidata":"https://www.wikidata.org/wiki/Q319913","display_name":"Convex function","level":3,"score":0.3138999938964844},{"id":"https://openalex.org/C49847556","wikidata":"https://www.wikidata.org/wiki/Q3964631","display_name":"Explained sum of squares","level":2,"score":0.2897000014781952},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.2879999876022339},{"id":"https://openalex.org/C182365436","wikidata":"https://www.wikidata.org/wiki/Q50701","display_name":"Variable (mathematics)","level":2,"score":0.2849999964237213},{"id":"https://openalex.org/C182419690","wikidata":"https://www.wikidata.org/wiki/Q205464","display_name":"Algebraic structure","level":2,"score":0.27619999647140503},{"id":"https://openalex.org/C157972887","wikidata":"https://www.wikidata.org/wiki/Q463359","display_name":"Convex optimization","level":3,"score":0.272599995136261},{"id":"https://openalex.org/C136979486","wikidata":"https://www.wikidata.org/wiki/Q773483","display_name":"Existential quantification","level":2,"score":0.2574000060558319},{"id":"https://openalex.org/C184264201","wikidata":"https://www.wikidata.org/wiki/Q21199","display_name":"Natural number","level":2,"score":0.2531000077724457}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1137/110856010","is_oa":false,"landing_page_url":"https://doi.org/10.1137/110856010","pdf_url":null,"source":{"id":"https://openalex.org/S928796702","display_name":"SIAM Journal on Optimization","issn_l":"1052-6234","issn":["1052-6234","1095-7189"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Optimization","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1111.4587","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1111.4587","pdf_url":"https://arxiv.org/pdf/1111.4587","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"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":"text"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:1111.4587","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1111.4587","pdf_url":"https://arxiv.org/pdf/1111.4587","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"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":"text"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":25,"referenced_works":["https://openalex.org/W1884347414","https://openalex.org/W1967184028","https://openalex.org/W1969120667","https://openalex.org/W1970515840","https://openalex.org/W1970556775","https://openalex.org/W1974962384","https://openalex.org/W1995488855","https://openalex.org/W2001334051","https://openalex.org/W2007264979","https://openalex.org/W2010553012","https://openalex.org/W2015144913","https://openalex.org/W2026578200","https://openalex.org/W2027111923","https://openalex.org/W2028123920","https://openalex.org/W2049087128","https://openalex.org/W2055993677","https://openalex.org/W2062200659","https://openalex.org/W2079135578","https://openalex.org/W2136991030","https://openalex.org/W2150929012","https://openalex.org/W2325692311","https://openalex.org/W2912522929","https://openalex.org/W2962725330","https://openalex.org/W4233252806","https://openalex.org/W4246855957"],"related_works":[],"abstract_inverted_index":{"Our":[0,128],"first":[1,29],"contribution":[2,85],"in":[3,70,138,167,173,181,194,203,225],"this":[4],"paper":[5],"is":[6,58,86,117,123,216],"to":[7,46,87,124],"prove":[8,88,162],"that":[9,89,131,217],"three":[10,40],"natural":[11],"sum":[12],"of":[13,21,26,65,73,135,141,149,171,234],"squares":[14],"(sos)":[15],"based":[16],"sufficient":[17],"conditions":[18],"for":[19,107,112,191,210],"convexity":[20,57],"polynomials,":[22],"via":[23],"the":[24,63,113,132,145,213],"definition":[25],"convexity,":[27],"its":[28,33],"order":[30,35],"characterization,":[31,36],"and":[32,67,78,93,122,153,176,179,184,187,206],"second":[34],"are":[37,222,232],"equivalent.":[38],"These":[39],"equivalent":[41],"algebraic":[42],"conditions,":[43],"henceforth":[44],"referred":[45],"as":[47,236],"sos-convexity,":[48],"can":[49],"be":[50,125],"checked":[51],"by":[52,238],"semidefinite":[53],"programming,":[54],"whereas":[55],"deciding":[56],"NP-hard.":[59],"If":[60],"we":[61,165],"denote":[62],"set":[64,133,146],"convex":[66,136,218],"sos-convex":[68,150,223],"polynomials":[69,172,219,229],"$n$":[71,139,204],"variables":[72,140,205],"degree":[74,142,207],"$d$":[75,143],"with":[76,120],"$\\tilde{C}_{n,d}$":[77],"$\\tilde{\\Sigma":[79],"C}_{n,d}$":[80,91],"respectively,":[81],"then":[82],"our":[83],"main":[84],"$\\tilde{C}_{n,d}=\\tilde{\\Sigma":[90],"if":[92,95,152,155],"only":[94,154],"$n=1$":[96],"or":[97,99,157,159],"$d=2$":[98,158],"$(n,d)=(2,4)$.":[100],"We":[101],"also":[102],"present":[103,166],"a":[104,188],"complete":[105],"characterization":[106],"forms":[108,137,151,180,193,202],"(homogeneous":[109],"polynomials)":[110],"except":[111],"case":[114],"$(n,d)=(3,4)$,":[115],"which":[116],"joint":[118],"work":[119],"Blekherman":[121],"published":[126],"elsewhere.":[127],"result":[129],"states":[130],"$C_{n,d}$":[134],"equals":[144],"$\\Sigma":[147],"C_{n,d}$":[148],"$n=2$":[156],"$(n,d)=(3,4)$.":[160],"To":[161],"these":[163],"results,":[164],"particular":[168],"explicit":[169],"examples":[170],"$\\tilde{C}_{2,6}\\setminus\\tilde{\\Sigma":[174],"C}_{2,6}$":[175],"$\\tilde{C}_{3,4}\\setminus\\tilde{\\Sigma":[177],"C}_{3,4}$":[178],"$C_{3,6}\\setminus\\Sigma":[182],"C_{3,6}$":[183],"$C_{4,4}\\setminus\\Sigma":[185],"C_{4,4,}$":[186],"general":[189],"procedure":[190],"constructing":[192],"$C_{n,d+2}\\setminus\\Sigma":[195],"C_{n,d+2}$":[196],"from":[197],"nonnegative":[198,228],"but":[199],"not":[200],"sos":[201],"$d$.":[208],"Although":[209],"disparate":[211],"reasons,":[212],"remarkable":[214],"outcome":[215],"(resp.,":[220,230],"forms)":[221,231],"exactly":[224],"cases":[226],"where":[227],"sums":[233],"squares,":[235],"characterized":[237],"Hilbert.":[239]},"counts_by_year":[{"year":2026,"cited_by_count":5},{"year":2025,"cited_by_count":6},{"year":2024,"cited_by_count":7},{"year":2023,"cited_by_count":4},{"year":2022,"cited_by_count":3},{"year":2021,"cited_by_count":6},{"year":2020,"cited_by_count":6},{"year":2019,"cited_by_count":4},{"year":2018,"cited_by_count":5},{"year":2017,"cited_by_count":5},{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":3},{"year":2014,"cited_by_count":3},{"year":2013,"cited_by_count":3}],"updated_date":"2026-06-18T10:00:31.954636","created_date":"2016-06-24T00:00:00"}
