{"id":"https://openalex.org/W7139989358","doi":"https://doi.org/10.1007/s11590-026-02289-7","title":"Convergence of the sum-of-squares hierarchy for quadratic optimization over roots-of-unity","display_name":"Convergence of the sum-of-squares hierarchy for quadratic optimization over roots-of-unity","publication_year":2026,"publication_date":"2026-03-21","ids":{"openalex":"https://openalex.org/W7139989358","doi":"https://doi.org/10.1007/s11590-026-02289-7"},"language":"en","primary_location":{"id":"doi:10.1007/s11590-026-02289-7","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s11590-026-02289-7","pdf_url":null,"source":{"id":"https://openalex.org/S12647387","display_name":"Optimization Letters","issn_l":"1862-4472","issn":["1862-4472","1862-4480"],"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":"Optimization Letters","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://doi.org/10.1007/s11590-026-02289-7","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5130262793","display_name":"Ahmad Al-Sulami","orcid":null},"institutions":[{"id":"https://openalex.org/I241749","display_name":"University of Cambridge","ror":"https://ror.org/013meh722","country_code":"GB","type":"education","lineage":["https://openalex.org/I241749"]}],"countries":["GB"],"is_corresponding":true,"raw_author_name":"Ahmad Al-Sulami","raw_affiliation_strings":["Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Rd, Cambridge, CB3 0WA, UK"],"raw_orcid":"https://orcid.org/0009-0001-3735-0579","affiliations":[{"raw_affiliation_string":"Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Rd, Cambridge, CB3 0WA, UK","institution_ids":["https://openalex.org/I241749"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5017261970","display_name":"Hamza Fawzi","orcid":null},"institutions":[{"id":"https://openalex.org/I241749","display_name":"University of Cambridge","ror":"https://ror.org/013meh722","country_code":"GB","type":"education","lineage":["https://openalex.org/I241749"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"Hamza Fawzi","raw_affiliation_strings":["Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Rd, Cambridge, CB3 0WA, UK"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Rd, Cambridge, CB3 0WA, UK","institution_ids":["https://openalex.org/I241749"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5016510897","display_name":"Shengding Sun","orcid":"https://orcid.org/0000-0002-9881-2618"},"institutions":[{"id":"https://openalex.org/I909854389","display_name":"London School of Economics and Political Science","ror":"https://ror.org/0090zs177","country_code":"GB","type":"education","lineage":["https://openalex.org/I124357947","https://openalex.org/I909854389"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"Shengding Sun","raw_affiliation_strings":["Department of Mathematics, London School of Economics, Houghton St, WC2A 2AE, London, UK"],"raw_orcid":"https://orcid.org/0000-0002-9881-2618","affiliations":[{"raw_affiliation_string":"Department of Mathematics, London School of Economics, Houghton St, WC2A 2AE, London, UK","institution_ids":["https://openalex.org/I909854389"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5130262793"],"corresponding_institution_ids":["https://openalex.org/I241749"],"apc_list":{"value":2890,"currency":"USD","value_usd":2890},"apc_paid":{"value":2890,"currency":"USD","value_usd":2890},"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.31888398,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"20","issue":"6","first_page":"1265","last_page":"1278"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.3068000078201294,"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"}},"topics":[{"id":"https://openalex.org/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.3068000078201294,"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/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.2092999964952469,"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/T12303","display_name":"Tensor decomposition and applications","score":0.12479999661445618,"subfield":{"id":"https://openalex.org/subfields/2605","display_name":"Computational Mathematics"},"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/hierarchy","display_name":"Hierarchy","score":0.5564000010490417},{"id":"https://openalex.org/keywords/quadratic-equation","display_name":"Quadratic equation","score":0.54339998960495},{"id":"https://openalex.org/keywords/generalization","display_name":"Generalization","score":0.5051000118255615},{"id":"https://openalex.org/keywords/binary-number","display_name":"Binary number","score":0.46000000834465027},{"id":"https://openalex.org/keywords/sequence","display_name":"Sequence (biology)","score":0.44269999861717224},{"id":"https://openalex.org/keywords/optimization-problem","display_name":"Optimization problem","score":0.4341000020503998},{"id":"https://openalex.org/keywords/quadratic-programming","display_name":"Quadratic programming","score":0.41850000619888306},{"id":"https://openalex.org/keywords/semidefinite-programming","display_name":"Semidefinite programming","score":0.41589999198913574}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7506999969482422},{"id":"https://openalex.org/C31170391","wikidata":"https://www.wikidata.org/wiki/Q188619","display_name":"Hierarchy","level":2,"score":0.5564000010490417},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.54339998960495},{"id":"https://openalex.org/C177148314","wikidata":"https://www.wikidata.org/wiki/Q170084","display_name":"Generalization","level":2,"score":0.5051000118255615},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.504800021648407},{"id":"https://openalex.org/C48372109","wikidata":"https://www.wikidata.org/wiki/Q3913","display_name":"Binary number","level":2,"score":0.46000000834465027},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.44269999861717224},{"id":"https://openalex.org/C137836250","wikidata":"https://www.wikidata.org/wiki/Q984063","display_name":"Optimization problem","level":2,"score":0.4341000020503998},{"id":"https://openalex.org/C81845259","wikidata":"https://www.wikidata.org/wiki/Q290117","display_name":"Quadratic programming","level":2,"score":0.41850000619888306},{"id":"https://openalex.org/C101901036","wikidata":"https://www.wikidata.org/wiki/Q2269096","display_name":"Semidefinite programming","level":2,"score":0.41589999198913574},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.40299999713897705},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.4004000127315521},{"id":"https://openalex.org/C94940","wikidata":"https://www.wikidata.org/wiki/Q652941","display_name":"Hermitian matrix","level":2,"score":0.38749998807907104},{"id":"https://openalex.org/C177179195","wikidata":"https://www.wikidata.org/wiki/Q7268372","display_name":"Quadratic unconstrained binary optimization","level":4,"score":0.37709999084472656},{"id":"https://openalex.org/C65148998","wikidata":"https://www.wikidata.org/wiki/Q7268373","display_name":"Quadratically constrained quadratic program","level":3,"score":0.37619999051094055},{"id":"https://openalex.org/C166437778","wikidata":"https://www.wikidata.org/wiki/Q50695","display_name":"Quadratic function","level":3,"score":0.3328000009059906},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3102000057697296},{"id":"https://openalex.org/C152022596","wikidata":"https://www.wikidata.org/wiki/Q7268361","display_name":"Quadratic form (statistics)","level":2,"score":0.30730000138282776},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.3066999912261963},{"id":"https://openalex.org/C79187972","wikidata":"https://www.wikidata.org/wiki/Q2654791","display_name":"Conic optimization","level":5,"score":0.3010999858379364},{"id":"https://openalex.org/C57869625","wikidata":"https://www.wikidata.org/wiki/Q1783502","display_name":"Rate of convergence","level":3,"score":0.29269999265670776},{"id":"https://openalex.org/C49712288","wikidata":"https://www.wikidata.org/wiki/Q77601250","display_name":"Positive-definite matrix","level":3,"score":0.262800008058548},{"id":"https://openalex.org/C98036226","wikidata":"https://www.wikidata.org/wiki/Q7268356","display_name":"Quadratic assignment problem","level":3,"score":0.2624000012874603},{"id":"https://openalex.org/C139502532","wikidata":"https://www.wikidata.org/wiki/Q1122090","display_name":"Computational intelligence","level":2,"score":0.25459998846054077}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1007/s11590-026-02289-7","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s11590-026-02289-7","pdf_url":null,"source":{"id":"https://openalex.org/S12647387","display_name":"Optimization Letters","issn_l":"1862-4472","issn":["1862-4472","1862-4480"],"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":"Optimization Letters","raw_type":"journal-article"},{"id":"pmh:oai:www.repository.cam.ac.uk:1810/405488","is_oa":true,"landing_page_url":"https://www.repository.cam.ac.uk/handle/1810/405488","pdf_url":null,"source":{"id":"https://openalex.org/S4306401777","display_name":"Apollo (University of Cambridge)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I241749","host_organization_name":"University of Cambridge","host_organization_lineage":["https://openalex.org/I241749"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Article"}],"best_oa_location":{"id":"doi:10.1007/s11590-026-02289-7","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s11590-026-02289-7","pdf_url":null,"source":{"id":"https://openalex.org/S12647387","display_name":"Optimization Letters","issn_l":"1862-4472","issn":["1862-4472","1862-4480"],"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":"Optimization Letters","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G1204051771","display_name":null,"funder_award_id":"EP/X032051/1","funder_id":"https://openalex.org/F4320334322","funder_display_name":"HORIZON EUROPE Framework Programme"}],"funders":[{"id":"https://openalex.org/F4320314707","display_name":"Government of the United Kingdom","ror":"https://ror.org/05wnh3t63"},{"id":"https://openalex.org/F4320314731","display_name":"UK Research and Innovation","ror":"https://ror.org/001aqnf71"},{"id":"https://openalex.org/F4320334322","display_name":"HORIZON EUROPE Framework Programme","ror":null},{"id":"https://openalex.org/F4320335743","display_name":"Saudi Arabian Cultural Mission","ror":null}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":37,"referenced_works":["https://openalex.org/W5795904","https://openalex.org/W41226580","https://openalex.org/W653291882","https://openalex.org/W1607580485","https://openalex.org/W1639981145","https://openalex.org/W1965511886","https://openalex.org/W1967344706","https://openalex.org/W1967790992","https://openalex.org/W1985123706","https://openalex.org/W2007104311","https://openalex.org/W2034769312","https://openalex.org/W2063046368","https://openalex.org/W2070596677","https://openalex.org/W2071550336","https://openalex.org/W2077397558","https://openalex.org/W2125240990","https://openalex.org/W2134737843","https://openalex.org/W2145080587","https://openalex.org/W2159739530","https://openalex.org/W2161367205","https://openalex.org/W2168417806","https://openalex.org/W2177435529","https://openalex.org/W2538025098","https://openalex.org/W2603258381","https://openalex.org/W2795779869","https://openalex.org/W2897469996","https://openalex.org/W2963694619","https://openalex.org/W3081982944","https://openalex.org/W3099419577","https://openalex.org/W3105711037","https://openalex.org/W3182160993","https://openalex.org/W3198872216","https://openalex.org/W4236826464","https://openalex.org/W4254696992","https://openalex.org/W4312092748","https://openalex.org/W4365815307","https://openalex.org/W4383213595"],"related_works":[],"abstract_inverted_index":{"Abstract":[0],"In":[1,138],"this":[2,84,139,143],"paper,":[3],"we":[4,141],"study":[5,79],"the":[6,13,37,69,75,80,99,118,119,133,148,167,171],"discrete":[7],"complex":[8,25],"quadratic":[9,21,39],"optimization":[10,42],"problem,":[11,85],"where":[12],"goal":[14],"is":[15,33,87,94,175],"to":[16,62,96,98],"maximize":[17],"a":[18,34,88],"multivariate":[19],"Hermitian":[20],"objective":[22],"function":[23],"over":[24],"m":[26],"\u2019th":[27],"roots":[28],"of":[29,36,71,74,90,110,132,181],"unity.":[30],"This":[31],"problem":[32,43],"generalization":[35],"well-studied":[38],"unconstrained":[40],"binary":[41,120],"(QUBO),":[44],"and":[45,56,113,145,174],"has":[46],"applications":[47,63],"in":[48,52,60,64,67,117,170],"signal":[49],"processing,":[50],"including":[51],"maximal":[53],"likelihood":[54],"estimation":[55],"multiple-input":[57],"multiple-output":[58],"detection,":[59],"addition":[61],"statistical":[65],"physics,":[66],"particular":[68],"computation":[70],"ground":[72],"states":[73],"Potts":[76],"model.":[77],"We":[78],"sum-of-squares":[81],"hierarchy":[82,134,149],"for":[83,158],"which":[86],"sequence":[89],"semidefinite":[91],"relaxations":[92],"that":[93,116,147],"known":[95],"converge":[97],"optimal":[100],"solution":[101],"after":[102,151],"at":[103,125,152],"most":[104,126,153],"n":[105],"steps.":[106],"A":[107],"previous":[108],"result":[109,144],"Fawzi,":[111],"Saunderson":[112],"Parrilo":[114],"showed":[115],"case":[121],"(":[122],"$$m=2$$":[123,172],"),":[124],"$$\\lceil":[127],"n/2":[128],"\\rceil":[129],"$$":[130],"levels":[131],"are":[135],"actually":[136],"needed.":[137],"paper":[140],"generalize":[142],"establish":[146],"converges":[150],"$$\\lfloor":[154],"n/2\\rfloor":[155],"+1$$":[156],"level":[157],"any":[159],"$$m":[160],"\\ge":[161],"2$$":[162],".":[163],"Our":[164],"proof":[165],"generalizes":[166],"ideas":[168],"used":[169],"case,":[173],"based":[176],"on":[177],"constructing":[178],"chordal":[179],"covers":[180],"appropriate":[182],"Cayley":[183],"graphs.":[184]},"counts_by_year":[],"updated_date":"2026-07-29T09:40:50.615796","created_date":"2026-03-22T00:00:00"}
