{"id":"https://openalex.org/W7151239709","doi":"https://doi.org/10.48550/arxiv.2604.04130","title":"Primal-Dual Methods for Nonsmooth Nonconvex Optimization with Orthogonality Constraints","display_name":"Primal-Dual Methods for Nonsmooth Nonconvex Optimization with Orthogonality Constraints","publication_year":2026,"publication_date":"2026-04-05","ids":{"openalex":"https://openalex.org/W7151239709","doi":"https://doi.org/10.48550/arxiv.2604.04130"},"language":null,"primary_location":{"id":"doi:10.48550/arxiv.2604.04130","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2604.04130","pdf_url":null,"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":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"article"},"type":"preprint","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.48550/arxiv.2604.04130","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5133105067","display_name":"Linglingzhi Zhu","orcid":null},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Zhu, Linglingzhi","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5133124239","display_name":"Wentao Ding","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Ding, Wentao","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5133104856","display_name":"Shangyuan Liu","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Liu, Shangyuan","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5133125166","display_name":"Anthony Man-Cho So","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"So, Anthony Man-Cho","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5133105067"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.9406999945640564,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.9406999945640564,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"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":0.031099999323487282,"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/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.009499999694526196,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/orthogonality","display_name":"Orthogonality","score":0.8260999917984009},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.6299999952316284},{"id":"https://openalex.org/keywords/augmented-lagrangian-method","display_name":"Augmented Lagrangian method","score":0.5839999914169312},{"id":"https://openalex.org/keywords/smoothing","display_name":"Smoothing","score":0.4943999946117401},{"id":"https://openalex.org/keywords/matching","display_name":"Matching (statistics)","score":0.48260000348091125},{"id":"https://openalex.org/keywords/optimization-problem","display_name":"Optimization problem","score":0.4278999865055084},{"id":"https://openalex.org/keywords/scalability","display_name":"Scalability","score":0.39079999923706055}],"concepts":[{"id":"https://openalex.org/C17137986","wikidata":"https://www.wikidata.org/wiki/Q215067","display_name":"Orthogonality","level":2,"score":0.8260999917984009},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6434999704360962},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.6360999941825867},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.6299999952316284},{"id":"https://openalex.org/C150452318","wikidata":"https://www.wikidata.org/wiki/Q4820432","display_name":"Augmented Lagrangian method","level":2,"score":0.5839999914169312},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5113999843597412},{"id":"https://openalex.org/C3770464","wikidata":"https://www.wikidata.org/wiki/Q775963","display_name":"Smoothing","level":2,"score":0.4943999946117401},{"id":"https://openalex.org/C165064840","wikidata":"https://www.wikidata.org/wiki/Q1321061","display_name":"Matching (statistics)","level":2,"score":0.48260000348091125},{"id":"https://openalex.org/C137836250","wikidata":"https://www.wikidata.org/wiki/Q984063","display_name":"Optimization problem","level":2,"score":0.4278999865055084},{"id":"https://openalex.org/C48044578","wikidata":"https://www.wikidata.org/wiki/Q727490","display_name":"Scalability","level":2,"score":0.39079999923706055},{"id":"https://openalex.org/C148043351","wikidata":"https://www.wikidata.org/wiki/Q4456944","display_name":"Current (fluid)","level":2,"score":0.3366999924182892},{"id":"https://openalex.org/C179799912","wikidata":"https://www.wikidata.org/wiki/Q205084","display_name":"Computational complexity theory","level":2,"score":0.3305000066757202},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.31690001487731934},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3149999976158142},{"id":"https://openalex.org/C55660270","wikidata":"https://www.wikidata.org/wiki/Q5164377","display_name":"Constrained optimization","level":2,"score":0.30149999260902405},{"id":"https://openalex.org/C2778029271","wikidata":"https://www.wikidata.org/wiki/Q5421931","display_name":"Extension (predicate logic)","level":2,"score":0.2709999978542328},{"id":"https://openalex.org/C48753275","wikidata":"https://www.wikidata.org/wiki/Q11216","display_name":"Numerical analysis","level":2,"score":0.2508000135421753},{"id":"https://openalex.org/C102634674","wikidata":"https://www.wikidata.org/wiki/Q868473","display_name":"Smoothness","level":2,"score":0.2500999867916107}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.48550/arxiv.2604.04130","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2604.04130","pdf_url":null,"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":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article"}],"best_oa_location":{"id":"doi:10.48550/arxiv.2604.04130","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2604.04130","pdf_url":null,"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":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"Recent":[0],"advancements":[1],"in":[2,118],"data":[3],"science":[4],"have":[5],"significantly":[6],"elevated":[7],"the":[8,29,36,40,115,119,126,137,151,158],"importance":[9],"of":[10,35,42,100,108,136,157],"orthogonally":[11],"constrained":[12,148],"optimization":[13,89,121],"problems.":[14],"The":[15],"Riemannian":[16,52,120],"approach":[17,74],"has":[18],"become":[19],"a":[20,71,77],"popular":[21],"technique":[22],"for":[23,85,110],"addressing":[24],"these":[25],"problems":[26,149],"due":[27],"to":[28,50],"advantageous":[30],"computational":[31,153],"and":[32,59,75,87,98,145,155],"analytical":[33],"properties":[34],"Stiefel":[37],"manifold.":[38],"Nonetheless,":[39],"interplay":[41],"nonsmoothness":[43],"alongside":[44],"orthogonality":[45,91],"constraints":[46],"introduces":[47],"substantial":[48],"challenges":[49],"current":[51],"methods,":[53],"including":[54],"scalability,":[55],"parallelizability,":[56],"complicated":[57],"subproblems,":[58],"cumulative":[60],"numerical":[61],"errors":[62],"that":[63],"threaten":[64],"feasibility.":[65],"In":[66],"this":[67],"paper,":[68],"we":[69,131],"take":[70],"retraction-free":[72],"primal-dual":[73],"propose":[76],"linearized":[78],"smoothing":[79],"augmented":[80],"Lagrangian":[81],"method":[82,95,160],"specifically":[83],"designed":[84],"nonsmooth":[86,146],"nonconvex":[88],"with":[90,162],"constraints.":[92],"Our":[93],"proposed":[94,138,159],"is":[96],"single-loop":[97],"free":[99],"subproblem":[101],"solving.":[102],"We":[103],"establish":[104],"its":[105],"iteration":[106],"complexity":[107],"$O(\u03b5^{-3})$":[109],"finding":[111],"$\u03b5$-KKT":[112],"points,":[113],"matching":[114],"best-known":[116],"results":[117],"literature.":[122],"Additionally,":[123],"by":[124],"invoking":[125],"standard":[127],"Kurdyka-Lojasiewicz":[128],"(KL)":[129],"property,":[130],"demonstrate":[132,150],"asymptotic":[133],"sequential":[134],"convergence":[135],"algorithm.":[139],"Numerical":[140],"experiments":[141],"on":[142],"both":[143],"smooth":[144],"orthogonal":[147],"superior":[152],"efficiency":[154],"scalability":[156],"compared":[161],"state-of-the-art":[163],"algorithms.":[164]},"counts_by_year":[],"updated_date":"2026-04-08T06:07:18.267832","created_date":"2026-04-08T00:00:00"}
