{"id":"https://openalex.org/W3203691100","doi":"https://doi.org/10.1137/21m1455127","title":"Convex Relaxations of Integral Variational Problems: Pointwise Dual Relaxation and Sum-of-Squares Optimization","display_name":"Convex Relaxations of Integral Variational Problems: Pointwise Dual Relaxation and Sum-of-Squares Optimization","publication_year":2023,"publication_date":"2023-05-19","ids":{"openalex":"https://openalex.org/W3203691100","doi":"https://doi.org/10.1137/21m1455127","mag":"3203691100"},"language":"en","primary_location":{"id":"doi:10.1137/21m1455127","is_oa":false,"landing_page_url":"https://doi.org/10.1137/21m1455127","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/2110.03079","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5040607634","display_name":"Alexander I. Chernyavsky","orcid":"https://orcid.org/0000-0002-3554-4599"},"institutions":[{"id":"https://openalex.org/I115441956","display_name":"Buffalo State University","ror":"https://ror.org/05ms04m92","country_code":"US","type":"education","lineage":["https://openalex.org/I115441956"]},{"id":"https://openalex.org/I212119943","display_name":"University of Victoria","ror":"https://ror.org/04s5mat29","country_code":"CA","type":"education","lineage":["https://openalex.org/I212119943"]},{"id":"https://openalex.org/I63190737","display_name":"University at Buffalo, State University of New York","ror":"https://ror.org/01y64my43","country_code":"US","type":"education","lineage":["https://openalex.org/I63190737"]}],"countries":["CA","US"],"is_corresponding":false,"raw_author_name":"Alexander Chernyavsky","raw_affiliation_strings":["Department of Mathematics and Statistics, University of Victoria, Victoria V8P 5C2, BC, Canada","Department of Mathematics, SUNY Buffalo, Buffalo, NY 14260 USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Statistics, University of Victoria, Victoria V8P 5C2, BC, Canada","institution_ids":["https://openalex.org/I212119943"]},{"raw_affiliation_string":"Department of Mathematics, SUNY Buffalo, Buffalo, NY 14260 USA","institution_ids":["https://openalex.org/I115441956","https://openalex.org/I63190737"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5017009058","display_name":"Jason J. Bramburger","orcid":"https://orcid.org/0000-0002-4687-4956"},"institutions":[{"id":"https://openalex.org/I212119943","display_name":"University of Victoria","ror":"https://ror.org/04s5mat29","country_code":"CA","type":"education","lineage":["https://openalex.org/I212119943"]},{"id":"https://openalex.org/I60158472","display_name":"Concordia University","ror":"https://ror.org/0420zvk78","country_code":"CA","type":"education","lineage":["https://openalex.org/I60158472"]}],"countries":["CA"],"is_corresponding":false,"raw_author_name":"Jason J. Bramburger","raw_affiliation_strings":["Department of Mathematics and Statistics, University of Victoria, Victoria V8P 5C2, BC, Canada","Mathematics and Statistics, Concordia University, Montreal H3G 1M8, QC, Canada"],"raw_orcid":"https://orcid.org/0000-0002-4687-4956","affiliations":[{"raw_affiliation_string":"Department of Mathematics and Statistics, University of Victoria, Victoria V8P 5C2, BC, Canada","institution_ids":["https://openalex.org/I212119943"]},{"raw_affiliation_string":"Mathematics and Statistics, Concordia University, Montreal H3G 1M8, QC, Canada","institution_ids":["https://openalex.org/I60158472"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5002663383","display_name":"Giovanni Fantuzzi","orcid":"https://orcid.org/0000-0002-0808-0944"},"institutions":[{"id":"https://openalex.org/I212119943","display_name":"University of Victoria","ror":"https://ror.org/04s5mat29","country_code":"CA","type":"education","lineage":["https://openalex.org/I212119943"]},{"id":"https://openalex.org/I47508984","display_name":"Imperial College London","ror":"https://ror.org/041kmwe10","country_code":"GB","type":"education","lineage":["https://openalex.org/I47508984"]},{"id":"https://openalex.org/I60158472","display_name":"Concordia University","ror":"https://ror.org/0420zvk78","country_code":"CA","type":"education","lineage":["https://openalex.org/I60158472"]}],"countries":["CA","GB"],"is_corresponding":false,"raw_author_name":"Giovanni Fantuzzi","raw_affiliation_strings":["Department of Aeronautics, Imperial College London, London SW7 2BX, UK","Mathematics and Statistics, Concordia University, Canada","Department of Mathematics and Statistics, University of Victoria, Canada"],"raw_orcid":"https://orcid.org/0000-0002-0808-0944","affiliations":[{"raw_affiliation_string":"Department of Aeronautics, Imperial College London, London SW7 2BX, UK","institution_ids":["https://openalex.org/I47508984"]},{"raw_affiliation_string":"Mathematics and Statistics, Concordia University, Canada","institution_ids":["https://openalex.org/I60158472"]},{"raw_affiliation_string":"Department of Mathematics and Statistics, University of Victoria, Canada","institution_ids":["https://openalex.org/I212119943"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5033695783","display_name":"David Goluskin","orcid":"https://orcid.org/0000-0003-3109-0830"},"institutions":[{"id":"https://openalex.org/I212119943","display_name":"University of Victoria","ror":"https://ror.org/04s5mat29","country_code":"CA","type":"education","lineage":["https://openalex.org/I212119943"]}],"countries":["CA"],"is_corresponding":false,"raw_author_name":"David Goluskin","raw_affiliation_strings":["Department of Mathematics and Statistics, University of Victoria, Victoria V8P 5C2, BC, Canada"],"raw_orcid":"https://orcid.org/0000-0003-3109-0830","affiliations":[{"raw_affiliation_string":"Department of Mathematics and Statistics, University of Victoria, Victoria V8P 5C2, BC, Canada","institution_ids":["https://openalex.org/I212119943"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":4,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":1.8965,"has_fulltext":false,"cited_by_count":5,"citation_normalized_percentile":{"value":0.81029743,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":98},"biblio":{"volume":"33","issue":"2","first_page":"481","last_page":"512"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.9994000196456909,"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":0.9994000196456909,"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/T10545","display_name":"Optimization and Variational Analysis","score":0.9847000241279602,"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/T10928","display_name":"Probabilistic and Robust Engineering Design","score":0.9742000102996826,"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/pointwise","display_name":"Pointwise","score":0.8848737478256226},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8602404594421387},{"id":"https://openalex.org/keywords/infimum-and-supremum","display_name":"Infimum and supremum","score":0.8054079413414001},{"id":"https://openalex.org/keywords/nabla-symbol","display_name":"Nabla symbol","score":0.61149662733078},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.5398684144020081},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.5150777697563171},{"id":"https://openalex.org/keywords/convex-function","display_name":"Convex function","score":0.46324148774147034},{"id":"https://openalex.org/keywords/relaxation","display_name":"Relaxation (psychology)","score":0.430949330329895},{"id":"https://openalex.org/keywords/omega","display_name":"Omega","score":0.3979852795600891},{"id":"https://openalex.org/keywords/regular-polygon","display_name":"Regular polygon","score":0.389191210269928},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.3563729524612427},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3528138995170593},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.290155827999115}],"concepts":[{"id":"https://openalex.org/C2777984123","wikidata":"https://www.wikidata.org/wiki/Q9248237","display_name":"Pointwise","level":2,"score":0.8848737478256226},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8602404594421387},{"id":"https://openalex.org/C95611797","wikidata":"https://www.wikidata.org/wiki/Q17502105","display_name":"Infimum and supremum","level":2,"score":0.8054079413414001},{"id":"https://openalex.org/C54207081","wikidata":"https://www.wikidata.org/wiki/Q937750","display_name":"Nabla symbol","level":3,"score":0.61149662733078},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5398684144020081},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.5150777697563171},{"id":"https://openalex.org/C145446738","wikidata":"https://www.wikidata.org/wiki/Q319913","display_name":"Convex function","level":3,"score":0.46324148774147034},{"id":"https://openalex.org/C2776029896","wikidata":"https://www.wikidata.org/wiki/Q3935810","display_name":"Relaxation (psychology)","level":2,"score":0.430949330329895},{"id":"https://openalex.org/C2779557605","wikidata":"https://www.wikidata.org/wiki/Q9890","display_name":"Omega","level":2,"score":0.3979852795600891},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.389191210269928},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3563729524612427},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3528138995170593},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.290155827999115},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C15744967","wikidata":"https://www.wikidata.org/wiki/Q9418","display_name":"Psychology","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/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C77805123","wikidata":"https://www.wikidata.org/wiki/Q161272","display_name":"Social psychology","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1137/21m1455127","is_oa":false,"landing_page_url":"https://doi.org/10.1137/21m1455127","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:2110.03079","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2110.03079","pdf_url":"https://arxiv.org/pdf/2110.03079","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:2110.03079","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2110.03079","pdf_url":"https://arxiv.org/pdf/2110.03079","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":[{"id":"https://openalex.org/G195882237","display_name":null,"funder_award_id":"RGPAS-2018-522657","funder_id":"https://openalex.org/F4320334593","funder_display_name":"Natural Sciences and Engineering Research Council of Canada"},{"id":"https://openalex.org/G3794971834","display_name":null,"funder_award_id":"DGECR-2018-0037","funder_id":"https://openalex.org/F4320334593","funder_display_name":"Natural Sciences and Engineering Research Council of Canada"}],"funders":[{"id":"https://openalex.org/F4320310984","display_name":"Pacific Institute for the Mathematical Sciences","ror":"https://ror.org/04w49dw51"},{"id":"https://openalex.org/F4320320283","display_name":"Imperial College London","ror":"https://ror.org/041kmwe10"},{"id":"https://openalex.org/F4320334593","display_name":"Natural Sciences and Engineering Research Council of Canada","ror":"https://ror.org/01h531d29"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":52,"referenced_works":["https://openalex.org/W404764422","https://openalex.org/W639861265","https://openalex.org/W1230513566","https://openalex.org/W1512048500","https://openalex.org/W1542648917","https://openalex.org/W1575147392","https://openalex.org/W1653619892","https://openalex.org/W1965076590","https://openalex.org/W1967184028","https://openalex.org/W1967344706","https://openalex.org/W2001532638","https://openalex.org/W2005630198","https://openalex.org/W2012648064","https://openalex.org/W2019852530","https://openalex.org/W2026578200","https://openalex.org/W2032231305","https://openalex.org/W2053179986","https://openalex.org/W2057170710","https://openalex.org/W2068371191","https://openalex.org/W2079135578","https://openalex.org/W2090394323","https://openalex.org/W2111145614","https://openalex.org/W2130458168","https://openalex.org/W2133404041","https://openalex.org/W2146586176","https://openalex.org/W2165281529","https://openalex.org/W2171586880","https://openalex.org/W2259408100","https://openalex.org/W2273822880","https://openalex.org/W2291124576","https://openalex.org/W2400408040","https://openalex.org/W2485934850","https://openalex.org/W2536620281","https://openalex.org/W2619881549","https://openalex.org/W2787397544","https://openalex.org/W2789205603","https://openalex.org/W2795744704","https://openalex.org/W2798718539","https://openalex.org/W2799231922","https://openalex.org/W2876051861","https://openalex.org/W2884208019","https://openalex.org/W2962716295","https://openalex.org/W2964252321","https://openalex.org/W2991334657","https://openalex.org/W3023172377","https://openalex.org/W3102481481","https://openalex.org/W3103708548","https://openalex.org/W3104314216","https://openalex.org/W3144358832","https://openalex.org/W4207084848","https://openalex.org/W4255442211","https://openalex.org/W4301021743"],"related_works":["https://openalex.org/W2229925025","https://openalex.org/W1539596337","https://openalex.org/W33469805","https://openalex.org/W1855007440","https://openalex.org/W2788247819","https://openalex.org/W2792440356","https://openalex.org/W3112745479","https://openalex.org/W4378474062","https://openalex.org/W1548053949","https://openalex.org/W4298324550"],"abstract_inverted_index":{"We":[0,57,176],"present":[1],"a":[2,59,73],"method":[3],"for":[4,189,240],"finding":[5],"lower":[6,74,181,225],"bounds":[7],"on":[8,76],"the":[9,65,77,80,92,112,124,147,179,185,219,223],"global":[10,50],"infima":[11],"of":[12,79,121,152,165,192,200,206,218,233],"integral":[13,47],"variational":[14],"problems,":[15,193],"wherein":[16],"$\\int_\\Omega":[17],"f(x,u(x),\\nabla":[18],"u(x)){\\rm":[19],"d}x$":[20],"is":[21,72,143,187,238],"minimized":[22],"over":[23,37,102,107],"functions":[24,120],"$u\\colon\\Omega\\subset\\mathbb{R}^n\\to\\mathbb{R}^m$":[25],"satisfying":[26],"given":[27,183],"equality":[28],"or":[29,39,44,100],"inequality":[30],"constraints.":[31],"Each":[32],"constraint":[33],"may":[34],"be":[35,86,116,127],"imposed":[36],"$\\Omega$":[38],"its":[40,95],"boundary,":[41],"either":[42],"pointwise":[43,66,98,132],"in":[45,236],"an":[46,163],"sense.":[48],"These":[49],"minimizations":[51],"are":[52,97,154,230],"generally":[53],"non-convex":[54],"and":[55,90,150,168,203],"intractable.":[56],"formulate":[58],"particular":[60],"convex":[61],"maximization,":[62],"here":[63,161],"called":[64],"dual":[67],"relaxation":[68],"(PDR),":[69],"whose":[70,194],"supremum":[71],"bound":[75,182,226],"infimum":[78],"original":[81,93,113],"problem.":[82],"The":[83,139,158],"PDR":[84,125,186,220],"can":[85,115,126],"derived":[87],"by":[88,118,130,184],"dualizing":[89],"relaxing":[91],"problem;":[94],"constraints":[96,153],"equalities":[99],"inequalities":[101,133],"finite-dimensional":[103],"sets,":[104],"rather":[105],"than":[106],"infinite-dimensional":[108],"function":[109],"spaces.":[110],"When":[111],"minimization":[114],"specified":[117],"polynomial":[119,135,228],"$(x,u,\\nabla":[122],"u)$,":[123],"further":[128],"relaxed":[129],"replacing":[131],"with":[134],"sum-of-squares":[136],"(SOS)":[137],"conditions.":[138],"resulting":[140],"SOS":[141,216,234],"program":[142],"computationally":[144],"tractable":[145],"when":[146],"dimensions":[148],"$m,n$":[149],"number":[151],"not":[155],"too":[156],"large.":[157],"framework":[159],"presented":[160],"generalizes":[162],"approach":[164],"Valmorbida,":[166],"Ahmadi,":[167],"Papachristodoulou":[169],"(IEEE":[170],"Trans.":[171],"Automat.":[172],"Contr.,":[173],"61:1649--1654,":[174],"2016).":[175],"prove":[177,214],"that":[178,215],"optimal":[180,204],"sharp":[188,224],"several":[190,241],"classes":[191],"special":[195],"cases":[196],"include":[197],"leading":[198],"eigenvalues":[199],"Sturm-Liouville":[201],"problems":[202],"constants":[205],"Poincar\\'e":[207],"inequalities.":[208],"For":[209],"these":[210],"same":[211],"classes,":[212],"we":[213],"relaxations":[217],"converge":[221],"to":[222],"as":[227],"degrees":[229],"increased.":[231],"Convergence":[232],"computations":[235],"practice":[237],"illustrated":[239],"examples.":[242]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":1}],"updated_date":"2026-06-22T08:00:12.763002","created_date":"2021-10-11T00:00:00"}
