{"id":"https://openalex.org/W4412781000","doi":"https://doi.org/10.1007/s10107-025-02235-y","title":"Second-order methods for quartically-regularised cubic polynomials, with applications to high-order tensor methods","display_name":"Second-order methods for quartically-regularised cubic polynomials, with applications to high-order tensor methods","publication_year":2025,"publication_date":"2025-07-31","ids":{"openalex":"https://openalex.org/W4412781000","doi":"https://doi.org/10.1007/s10107-025-02235-y"},"language":"en","primary_location":{"id":"doi:10.1007/s10107-025-02235-y","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10107-025-02235-y","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10107-025-02235-y.pdf","source":{"id":"https://openalex.org/S193920097","display_name":"Mathematical Programming","issn_l":"0025-5610","issn":["0025-5610","1436-4646"],"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":"Mathematical Programming","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s10107-025-02235-y.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5064170535","display_name":"Coralia Cartis","orcid":"https://orcid.org/0000-0002-0963-5550"},"institutions":[{"id":"https://openalex.org/I40120149","display_name":"University of Oxford","ror":"https://ror.org/052gg0110","country_code":"GB","type":"education","lineage":["https://openalex.org/I40120149"]}],"countries":["GB"],"is_corresponding":true,"raw_author_name":"Coralia Cartis","raw_affiliation_strings":["Mathematical Institute, University of Oxford, Woodstock Road, Oxford, OX2 6GG, UK"],"raw_orcid":"https://orcid.org/0000-0002-0963-5550","affiliations":[{"raw_affiliation_string":"Mathematical Institute, University of Oxford, Woodstock Road, Oxford, OX2 6GG, UK","institution_ids":["https://openalex.org/I40120149"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5100301546","display_name":"Wenqi Zhu","orcid":"https://orcid.org/0009-0008-0357-441X"},"institutions":[{"id":"https://openalex.org/I40120149","display_name":"University of Oxford","ror":"https://ror.org/052gg0110","country_code":"GB","type":"education","lineage":["https://openalex.org/I40120149"]}],"countries":["GB"],"is_corresponding":true,"raw_author_name":"Wenqi Zhu","raw_affiliation_strings":["Mathematical Institute, University of Oxford, Woodstock Road, Oxford, OX2 6GG, UK"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Mathematical Institute, University of Oxford, Woodstock Road, Oxford, OX2 6GG, UK","institution_ids":["https://openalex.org/I40120149"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5064170535","https://openalex.org/A5100301546"],"corresponding_institution_ids":["https://openalex.org/I40120149"],"apc_list":{"value":2190,"currency":"EUR","value_usd":2890},"apc_paid":{"value":2190,"currency":"EUR","value_usd":2890},"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.25433898,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"215","issue":"1-2","first_page":"669","last_page":"715"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.9994999766349792,"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":0.9994999766349792,"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/T10792","display_name":"Matrix Theory and Algorithms","score":0.9977999925613403,"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/T12303","display_name":"Tensor decomposition and applications","score":0.9973999857902527,"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/mathematics","display_name":"Mathematics","score":0.8039801716804504},{"id":"https://openalex.org/keywords/order","display_name":"Order (exchange)","score":0.593212366104126},{"id":"https://openalex.org/keywords/tensor","display_name":"Tensor (intrinsic definition)","score":0.5385021567344666},{"id":"https://openalex.org/keywords/numerical-analysis","display_name":"Numerical analysis","score":0.5102527141571045},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.46367642283439636},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.3446457087993622},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.3204777240753174},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.3002334237098694}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8039801716804504},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.593212366104126},{"id":"https://openalex.org/C155281189","wikidata":"https://www.wikidata.org/wiki/Q3518150","display_name":"Tensor (intrinsic definition)","level":2,"score":0.5385021567344666},{"id":"https://openalex.org/C48753275","wikidata":"https://www.wikidata.org/wiki/Q11216","display_name":"Numerical analysis","level":2,"score":0.5102527141571045},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.46367642283439636},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.3446457087993622},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3204777240753174},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3002334237098694},{"id":"https://openalex.org/C10138342","wikidata":"https://www.wikidata.org/wiki/Q43015","display_name":"Finance","level":1,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1007/s10107-025-02235-y","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10107-025-02235-y","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10107-025-02235-y.pdf","source":{"id":"https://openalex.org/S193920097","display_name":"Mathematical Programming","issn_l":"0025-5610","issn":["0025-5610","1436-4646"],"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":"Mathematical Programming","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.1007/s10107-025-02235-y","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10107-025-02235-y","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10107-025-02235-y.pdf","source":{"id":"https://openalex.org/S193920097","display_name":"Mathematical Programming","issn_l":"0025-5610","issn":["0025-5610","1436-4646"],"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":"Mathematical Programming","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320321920","display_name":"Innovation and Technology Commission","ror":"https://ror.org/04vf9tr09"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4412781000.pdf","grobid_xml":"https://content.openalex.org/works/W4412781000.grobid-xml"},"referenced_works_count":37,"referenced_works":["https://openalex.org/W404764422","https://openalex.org/W1499399028","https://openalex.org/W1568816820","https://openalex.org/W1967344706","https://openalex.org/W1972148564","https://openalex.org/W1991638696","https://openalex.org/W2000825868","https://openalex.org/W2004357692","https://openalex.org/W2009941369","https://openalex.org/W2012604196","https://openalex.org/W2042032299","https://openalex.org/W2042882259","https://openalex.org/W2056603712","https://openalex.org/W2112341314","https://openalex.org/W2132337079","https://openalex.org/W2149454052","https://openalex.org/W2154904851","https://openalex.org/W2156005216","https://openalex.org/W2159503542","https://openalex.org/W2165281529","https://openalex.org/W2510806995","https://openalex.org/W2530585400","https://openalex.org/W2558881672","https://openalex.org/W2619018971","https://openalex.org/W2808862920","https://openalex.org/W2962948705","https://openalex.org/W2980794445","https://openalex.org/W2982216356","https://openalex.org/W3040026369","https://openalex.org/W3092425772","https://openalex.org/W3092479710","https://openalex.org/W3212136124","https://openalex.org/W4220880548","https://openalex.org/W4313052657","https://openalex.org/W4400268749","https://openalex.org/W6600788769","https://openalex.org/W6638380949"],"related_works":["https://openalex.org/W3214346900","https://openalex.org/W2356099978","https://openalex.org/W3202709773","https://openalex.org/W2012593053","https://openalex.org/W1756394533","https://openalex.org/W1977848887","https://openalex.org/W3129086539","https://openalex.org/W2347719368","https://openalex.org/W2363346090","https://openalex.org/W4285308861"],"abstract_inverted_index":{"Abstract":[0],"There":[1],"has":[2,56],"been":[3,57],"growing":[4],"interest":[5],"in":[6,145,152,161,196],"high-order":[7],"tensor":[8,112],"methods":[9],"for":[10,46,71,191],"nonconvex":[11,35,74,198],"optimization,":[12],"with":[13,91,237,248],"adaptive":[14],"regularization,":[15],"as":[16,79,241,245],"they":[17],"possess":[18],"better/optimal":[19],"worst-case":[20],"evaluation":[21],"complexity":[22],"globally":[23],"and":[24,120],"faster":[25],"convergence":[26],"asymptotically.":[27],"These":[28],"algorithms":[29],"crucially":[30],"rely":[31],"on":[32,218],"repeatedly":[33],"minimizing":[34,73],"multivariate":[36],"Taylor-based":[37],"polynomial":[38],"sub-problems,":[39,51],"at":[40,170,201],"least":[41,171,202],"locally.":[42],"Finding":[43],"efficient":[44],"techniques":[45],"the":[47,53,80,110,146,150,159,162,197,223],"solution":[48],"of":[49,118,149,221],"these":[50,219],"beyond":[52],"second-order":[54,65],"case,":[55,199],"an":[58],"open":[59],"question.":[60],"This":[61],"paper":[62],"proposes":[63],"a":[64,115],"method,":[66],"Quadratic":[67],"Quartic":[68],"Regularisation":[69],"(QQR),":[70],"efficiently":[72],"quartically-regularized":[75],"cubic":[76],"polynomials,":[77],"such":[78,240],"AR":[81,246],"p":[82,247],"sub-problem":[83,151],"(Birgin":[84],"et":[85],"al.":[86],"Math":[87],"Program":[88],"163:359\u2013368,":[89],"2017)":[90],"$$p=3$$":[92],"<mml:math":[93,141,175,206,250],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">":[94,142,176,207,251],"<mml:mrow>":[95,177,182,208,252],"<mml:mi>p</mml:mi>":[96,253],"<mml:mo>=</mml:mo>":[97,254],"<mml:mn>3</mml:mn>":[98,185],"</mml:mrow>":[99,186,189,213,256],"</mml:math>":[100,144,190,214,257],".":[101],"Inspired":[102],"by":[103,114,169,200],"Nesterov":[104],"(Quartic":[105],"regularity,":[106],"2022),":[107],"QQR":[108,163,233],"approximates":[109],"third-order":[111],"term":[113],"linear":[116],"combination":[117],"quadratic":[119],"quartic":[121],"terms,":[122],"yielding":[123],"(possibly":[124],"nonconvex)":[125],"local":[126],"models":[127],"that":[128,158,231],"are":[129],"solvable":[130],"to":[131,136],"global":[132],"optimality.":[133],"In":[134],"order":[135],"achieve":[137],"accuracy":[138],"$$\\epsilon":[139],"$$":[140],"<mml:mi>\u03f5</mml:mi>":[143,181,211],"first-order":[147],"criticality":[148],"finitely":[153],"many":[154],"iterations,":[155,194,222],"we":[156],"show":[157],"error":[160],"method":[164],"decreases":[165],"either":[166,260],"linearly":[167],"or":[168,264],"$$\\mathcal":[172,203],"{O}(\\epsilon":[173,204],"^{4/3})$$":[174],"<mml:mi>O</mml:mi>":[178,209],"<mml:mo>(</mml:mo>":[179,210],"<mml:msup>":[180],"<mml:mn>4</mml:mn>":[183],"<mml:mo>/</mml:mo>":[184],"</mml:msup>":[187],"<mml:mo>)</mml:mo>":[188,212],"locally":[192],"convex":[193],"while":[195],")$$":[205],";":[215],"thus":[216],"improving,":[217],"types":[220],"general":[224],"cubic-regularization":[225],"bound.":[226],"Preliminary":[227],"numerical":[228],"experiments":[229],"indicate":[230],"two":[232],"variants":[234],"perform":[235],"competitively":[236],"state-of-the-art":[238],"approaches":[239],"ARC":[242],"(also":[243],"known":[244],"$$p=2$$":[249],"<mml:mn>2</mml:mn>":[255],"),":[258],"achieving":[259],"lower":[261],"objective":[262],"value":[263],"iteration":[265],"counts.":[266]},"counts_by_year":[],"updated_date":"2026-06-13T06:13:01.061226","created_date":"2025-10-10T00:00:00"}