{"id":"https://openalex.org/W7213523973","doi":"https://doi.org/10.48550/arxiv.2609.17780","title":"The Complexity of Finding Stationary Points in Nonsmooth Nonconvex Optimization","display_name":"The Complexity of Finding Stationary Points in Nonsmooth Nonconvex Optimization","publication_year":2026,"publication_date":"2026-09-15","ids":{"openalex":"https://openalex.org/W7213523973","doi":"https://doi.org/10.48550/arxiv.2609.17780"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.17780","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.17780","pdf_url":"https://arxiv.org/pdf/2609.17780","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2609.17780","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5064469637","display_name":"Guy Kornowski","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Kornowski, Guy","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5064469637"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":2,"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":0.9560999870300293,"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":0.9560999870300293,"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":0.025800000876188278,"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":0.010700000450015068,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/nonsmooth-nonconvex-optimization","display_name":"nonsmooth nonconvex optimization","score":0.9750000238418579},{"id":"https://openalex.org/keywords/lower-bounds","display_name":"lower bounds","score":0.8529999852180481},{"id":"https://openalex.org/keywords/first-order-algorithms","display_name":"first-order algorithms","score":0.8830000162124634},{"id":"https://openalex.org/keywords/lipschitz-functions","display_name":"Lipschitz functions","score":0.7950000166893005},{"id":"https://openalex.org/keywords/convergence-rate","display_name":"convergence rate","score":0.6830000281333923}],"concepts":[{"id":"https://openalex.org/C22324862","wikidata":"https://www.wikidata.org/wiki/Q652707","display_name":"Lipschitz continuity","level":2,"score":0.7746999859809875},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7653999924659729},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.5792999863624573},{"id":"https://openalex.org/C189237950","wikidata":"https://www.wikidata.org/wiki/Q2500758","display_name":"Stationary point","level":2,"score":0.527400016784668},{"id":"https://openalex.org/C191795146","wikidata":"https://www.wikidata.org/wiki/Q3878446","display_name":"Norm (philosophy)","level":2,"score":0.524399995803833},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.5131999850273132},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.5019000172615051},{"id":"https://openalex.org/C165064840","wikidata":"https://www.wikidata.org/wiki/Q1321061","display_name":"Matching (statistics)","level":2,"score":0.4708999991416931},{"id":"https://openalex.org/C153258448","wikidata":"https://www.wikidata.org/wiki/Q1199743","display_name":"Gradient descent","level":3,"score":0.4041000008583069},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.4000999927520752},{"id":"https://openalex.org/C28719098","wikidata":"https://www.wikidata.org/wiki/Q44946","display_name":"Point (geometry)","level":2,"score":0.3946000039577484},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.376800000667572},{"id":"https://openalex.org/C157972887","wikidata":"https://www.wikidata.org/wiki/Q463359","display_name":"Convex optimization","level":3,"score":0.35910001397132874},{"id":"https://openalex.org/C2776502983","wikidata":"https://www.wikidata.org/wiki/Q690182","display_name":"Contrast (vision)","level":2,"score":0.35499998927116394},{"id":"https://openalex.org/C10494615","wikidata":"https://www.wikidata.org/wiki/Q17086765","display_name":"Proximal Gradient Methods","level":4,"score":0.3488999903202057},{"id":"https://openalex.org/C145446738","wikidata":"https://www.wikidata.org/wiki/Q319913","display_name":"Convex function","level":3,"score":0.3260999917984009},{"id":"https://openalex.org/C57869625","wikidata":"https://www.wikidata.org/wiki/Q1783502","display_name":"Rate of convergence","level":3,"score":0.325300008058548},{"id":"https://openalex.org/C115680565","wikidata":"https://www.wikidata.org/wiki/Q5977448","display_name":"Gradient method","level":2,"score":0.2973000109195709},{"id":"https://openalex.org/C137836250","wikidata":"https://www.wikidata.org/wiki/Q984063","display_name":"Optimization problem","level":2,"score":0.2856999933719635},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.26460000872612},{"id":"https://openalex.org/C179799912","wikidata":"https://www.wikidata.org/wiki/Q205084","display_name":"Computational complexity theory","level":2,"score":0.25429999828338623}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.17780","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.17780","pdf_url":"https://arxiv.org/pdf/2609.17780","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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"},{"id":"doi:10.48550/arxiv.2609.17780","doi":"https://doi.org/10.48550/arxiv.2609.17780","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.17780","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.17780","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.17780","pdf_url":"https://arxiv.org/pdf/2609.17780","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7213523973.pdf","grobid_xml":"https://content.openalex.org/works/W7213523973.grobid-xml"},"referenced_works_count":5,"referenced_works":["https://openalex.org/W7155539384","https://openalex.org/W7171494670","https://openalex.org/W7171525360","https://openalex.org/W7172487772","https://openalex.org/W7204546023"],"related_works":["https://openalex.org/W4297434938","https://openalex.org/W4297899866","https://openalex.org/W7214033371","https://openalex.org/W7221532040","https://openalex.org/W4383993447","https://openalex.org/W2765150911","https://openalex.org/W2993258424","https://openalex.org/W2766533971","https://openalex.org/W3008792859","https://openalex.org/W3005972568"],"abstract_inverted_index":{"We":[0,65],"prove":[1,67],"that":[2,93],"first-order":[3],"algorithms":[4,48],"require":[5],"$\u03a9(\u03b4^{-1}\u03b5^{-3})$":[6],"gradient":[7,103],"queries":[8],"(in":[9],"the":[10,55,78],"worst":[11],"case)":[12],"to":[13,50,59,96,108],"find":[14],"a":[15,20,27,68],"$(\u03b4,\u03b5)$-Goldstein":[16],"stationary":[17],"point":[18],"of":[19,30,57,72,83],"Lipschitz":[21],"function,":[22],"at":[23,38],"which":[24,85],"there":[25],"is":[26,37,44],"convex":[28],"combination":[29],"gradients":[31],"within":[32],"distance":[33],"$\u03b4$":[34],"whose":[35],"norm":[36],"most":[39],"$\u03b5$.":[40],"This":[41],"lower":[42,70],"bound":[43,71],"tight,":[45],"matching":[46],"known":[47],"up":[49],"absolute":[51],"constants,":[52],"therefore":[53],"resolving":[54],"complexity":[56],"convergence":[58,94],"stationarity":[60,98],"in":[61,105],"nonsmooth":[62,97],"nonconvex":[63],"optimization.":[64,110],"further":[66],"tight":[69],"$\u03a9(\u03bb^{1/2}\u03b5^{-7/2})$":[73],"for":[74],"finding":[75],"points":[76],"satisfying":[77],"recently":[79],"proposed":[80],"relaxed":[81],"notion":[82],"$(\u03bb,\u03b5)$-stationarity,":[84],"allows":[86],"combining":[87],"further-away":[88],"gradients.":[89],"Our":[90],"results":[91],"reveal":[92],"rates":[95],"are":[99],"not":[100],"affected":[101],"by":[102],"stochasticity,":[104],"sharp":[106],"contrast":[107],"smooth":[109]},"counts_by_year":[{"year":2026,"cited_by_count":2}],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-18T00:00:00"}