{"id":"https://openalex.org/W2997080082","doi":"https://doi.org/10.1007/s10589-020-00259-y","title":"Stochastic proximal gradient methods for nonconvex problems in Hilbert spaces","display_name":"Stochastic proximal gradient methods for nonconvex problems in Hilbert spaces","publication_year":2021,"publication_date":"2021-01-12","ids":{"openalex":"https://openalex.org/W2997080082","doi":"https://doi.org/10.1007/s10589-020-00259-y","mag":"2997080082","pmid":"https://pubmed.ncbi.nlm.nih.gov/33707813"},"language":"en","primary_location":{"id":"doi:10.1007/s10589-020-00259-y","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10589-020-00259-y","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10589-020-00259-y.pdf","source":{"id":"https://openalex.org/S39243548","display_name":"Computational Optimization and Applications","issn_l":"0926-6003","issn":["0926-6003","1573-2894"],"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":"Computational Optimization and Applications","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref","pubmed"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s10589-020-00259-y.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":null,"display_name":"Caroline Geiersbach","orcid":null},"institutions":[{"id":"https://openalex.org/I4210092021","display_name":"Weierstrass Institute for Applied Analysis and Stochastics","ror":"https://ror.org/00h1x4t21","country_code":"DE","type":"facility","lineage":["https://openalex.org/I315704651","https://openalex.org/I4210092021"]}],"countries":["DE"],"is_corresponding":true,"raw_author_name":"Caroline Geiersbach","raw_affiliation_strings":["Weierstrass Institute, Mohrenstrasse 39, 10117 Berlin, Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Weierstrass Institute, Mohrenstrasse 39, 10117 Berlin, Germany","institution_ids":["https://openalex.org/I4210092021"]}]},{"author_position":"last","author":{"id":null,"display_name":"Teresa Scarinci","orcid":null},"institutions":[{"id":"https://openalex.org/I26415053","display_name":"University of L'Aquila","ror":"https://ror.org/01j9p1r26","country_code":"IT","type":"education","lineage":["https://openalex.org/I26415053"]}],"countries":["IT"],"is_corresponding":true,"raw_author_name":"Teresa Scarinci","raw_affiliation_strings":["Department of Information Engineering, Computer Science and Mathematics, University of L'Aquila, Via Vetoio - Loc. Coppito, 67010 L'Aquila, Italy"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Information Engineering, Computer Science and Mathematics, University of L'Aquila, Via Vetoio - Loc. Coppito, 67010 L'Aquila, Italy","institution_ids":["https://openalex.org/I26415053"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I26415053","https://openalex.org/I4210092021"],"apc_list":{"value":2990,"currency":"USD","value_usd":2990},"apc_paid":{"value":2990,"currency":"USD","value_usd":2990},"fwci":2.0101,"has_fulltext":true,"cited_by_count":16,"citation_normalized_percentile":{"value":0.87997868,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":91,"max":98},"biblio":{"volume":"78","issue":"3","first_page":"705","last_page":"740"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.9472000002861023,"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":0.9472000002861023,"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/T10545","display_name":"Optimization and Variational Analysis","score":0.016599999740719795,"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/T11413","display_name":"Risk and Portfolio Optimization","score":0.012199999764561653,"subfield":{"id":"https://openalex.org/subfields/1803","display_name":"Management Science and Operations Research"},"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/lipschitz-continuity","display_name":"Lipschitz continuity","score":0.7502999901771545},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.6071000099182129},{"id":"https://openalex.org/keywords/hilbert-space","display_name":"Hilbert space","score":0.5814999938011169},{"id":"https://openalex.org/keywords/stochastic-optimization","display_name":"Stochastic optimization","score":0.5629000067710876},{"id":"https://openalex.org/keywords/sequence","display_name":"Sequence (biology)","score":0.5220000147819519},{"id":"https://openalex.org/keywords/weak-convergence","display_name":"Weak convergence","score":0.4975999891757965},{"id":"https://openalex.org/keywords/convergence-of-random-variables","display_name":"Convergence of random variables","score":0.47200000286102295},{"id":"https://openalex.org/keywords/stationary-point","display_name":"Stationary point","score":0.46239998936653137},{"id":"https://openalex.org/keywords/proximal-gradient-methods","display_name":"Proximal Gradient Methods","score":0.46070000529289246}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8568999767303467},{"id":"https://openalex.org/C22324862","wikidata":"https://www.wikidata.org/wiki/Q652707","display_name":"Lipschitz continuity","level":2,"score":0.7502999901771545},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.6071000099182129},{"id":"https://openalex.org/C62799726","wikidata":"https://www.wikidata.org/wiki/Q190056","display_name":"Hilbert space","level":2,"score":0.5814999938011169},{"id":"https://openalex.org/C194387892","wikidata":"https://www.wikidata.org/wiki/Q1747770","display_name":"Stochastic optimization","level":2,"score":0.5629000067710876},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.53329998254776},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.5220000147819519},{"id":"https://openalex.org/C57945734","wikidata":"https://www.wikidata.org/wiki/Q7977941","display_name":"Weak convergence","level":3,"score":0.4975999891757965},{"id":"https://openalex.org/C95763700","wikidata":"https://www.wikidata.org/wiki/Q578985","display_name":"Convergence of random variables","level":3,"score":0.47200000286102295},{"id":"https://openalex.org/C189237950","wikidata":"https://www.wikidata.org/wiki/Q2500758","display_name":"Stationary point","level":2,"score":0.46239998936653137},{"id":"https://openalex.org/C10494615","wikidata":"https://www.wikidata.org/wiki/Q17086765","display_name":"Proximal Gradient Methods","level":4,"score":0.46070000529289246},{"id":"https://openalex.org/C63548660","wikidata":"https://www.wikidata.org/wiki/Q858223","display_name":"Limit point","level":2,"score":0.4505000114440918},{"id":"https://openalex.org/C151201525","wikidata":"https://www.wikidata.org/wiki/Q177239","display_name":"Limit (mathematics)","level":2,"score":0.44760000705718994},{"id":"https://openalex.org/C137836250","wikidata":"https://www.wikidata.org/wiki/Q984063","display_name":"Optimization problem","level":2,"score":0.43220001459121704},{"id":"https://openalex.org/C55479107","wikidata":"https://www.wikidata.org/wiki/Q97663916","display_name":"Stochastic approximation","level":3,"score":0.4260999858379364},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.39629998803138733},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.39419999718666077},{"id":"https://openalex.org/C51955184","wikidata":"https://www.wikidata.org/wiki/Q1545585","display_name":"Stochastic differential equation","level":2,"score":0.3928999900817871},{"id":"https://openalex.org/C157972887","wikidata":"https://www.wikidata.org/wiki/Q463359","display_name":"Convex optimization","level":3,"score":0.3896999955177307},{"id":"https://openalex.org/C145446738","wikidata":"https://www.wikidata.org/wiki/Q319913","display_name":"Convex function","level":3,"score":0.3790999948978424},{"id":"https://openalex.org/C122123141","wikidata":"https://www.wikidata.org/wiki/Q176623","display_name":"Random variable","level":2,"score":0.3783000111579895},{"id":"https://openalex.org/C115680565","wikidata":"https://www.wikidata.org/wiki/Q5977448","display_name":"Gradient method","level":2,"score":0.366100013256073},{"id":"https://openalex.org/C8272713","wikidata":"https://www.wikidata.org/wiki/Q176737","display_name":"Stochastic process","level":2,"score":0.3474000096321106},{"id":"https://openalex.org/C109578324","wikidata":"https://www.wikidata.org/wiki/Q3354463","display_name":"Random optimization","level":5,"score":0.32499998807907104},{"id":"https://openalex.org/C206688291","wikidata":"https://www.wikidata.org/wiki/Q7617819","display_name":"Stochastic gradient descent","level":3,"score":0.3215000033378601},{"id":"https://openalex.org/C61797465","wikidata":"https://www.wikidata.org/wiki/Q1188986","display_name":"Term (time)","level":2,"score":0.32120001316070557},{"id":"https://openalex.org/C71287196","wikidata":"https://www.wikidata.org/wiki/Q847204","display_name":"Limit of a sequence","level":3,"score":0.28610000014305115},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.25780001282691956}],"mesh":[],"locations_count":6,"locations":[{"id":"doi:10.1007/s10589-020-00259-y","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10589-020-00259-y","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10589-020-00259-y.pdf","source":{"id":"https://openalex.org/S39243548","display_name":"Computational Optimization and Applications","issn_l":"0926-6003","issn":["0926-6003","1573-2894"],"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":"Computational Optimization and Applications","raw_type":"journal-article"},{"id":"pmid:33707813","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/33707813","pdf_url":null,"source":{"id":"https://openalex.org/S4306525036","display_name":"PubMed","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Computational optimization and applications","raw_type":null},{"id":"pmh:oai:arXiv.org:2001.01329","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2001.01329","pdf_url":"https://arxiv.org/pdf/2001.01329","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,"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":"pmh:oai:RePEc:spr:coopap:v:78:y:2021:i:3:d:10.1007_s10589-020-00259-y","is_oa":false,"landing_page_url":"http://link.springer.com/10.1007/s10589-020-00259-y","pdf_url":null,"source":{"id":"https://openalex.org/S4306401271","display_name":"RePEc: Research Papers in Economics","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I77793887","host_organization_name":"Federal Reserve Bank of St. Louis","host_organization_lineage":["https://openalex.org/I77793887"],"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":"article"},{"id":"pmh:oai:econstor.eu:10419/287443","is_oa":true,"landing_page_url":"https://hdl.handle.net/10419/287443","pdf_url":null,"source":{"id":"https://openalex.org/S4306401696","display_name":"Econstor (Econstor)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"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":"","raw_type":"doc-type:article"},{"id":"pmh:oai:pubmedcentral.nih.gov:7907055","is_oa":true,"landing_page_url":"https://www.ncbi.nlm.nih.gov/pmc/articles/7907055","pdf_url":null,"source":{"id":"https://openalex.org/S2764455111","display_name":"PubMed Central","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"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":"Comput Optim Appl","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.1007/s10589-020-00259-y","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10589-020-00259-y","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10589-020-00259-y.pdf","source":{"id":"https://openalex.org/S39243548","display_name":"Computational Optimization and Applications","issn_l":"0926-6003","issn":["0926-6003","1573-2894"],"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":"Computational Optimization and Applications","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G6389310286","display_name":"Vienna Graduate School on Computational Optimization","funder_award_id":"W 1260","funder_id":"https://openalex.org/F4320321181","funder_display_name":"Austrian Science Fund"}],"funders":[{"id":"https://openalex.org/F4320311030","display_name":"Istituto Nazionale di Alta Matematica \"Francesco Severi\"","ror":"https://ror.org/01vx64p53"},{"id":"https://openalex.org/F4320321181","display_name":"Austrian Science Fund","ror":"https://ror.org/013tf3c58"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2997080082.pdf","grobid_xml":"https://content.openalex.org/works/W2997080082.grobid-xml"},"referenced_works_count":45,"referenced_works":["https://openalex.org/W596189563","https://openalex.org/W1496662771","https://openalex.org/W1551360398","https://openalex.org/W1976414317","https://openalex.org/W1987086517","https://openalex.org/W1992208280","https://openalex.org/W1994616650","https://openalex.org/W2001407827","https://openalex.org/W2007876740","https://openalex.org/W2009797711","https://openalex.org/W2011007482","https://openalex.org/W2022176369","https://openalex.org/W2029463628","https://openalex.org/W2038084402","https://openalex.org/W2040960244","https://openalex.org/W2043153039","https://openalex.org/W2053670647","https://openalex.org/W2060471940","https://openalex.org/W2067780150","https://openalex.org/W2079137634","https://openalex.org/W2080364619","https://openalex.org/W2119374337","https://openalex.org/W2123406850","https://openalex.org/W2125812768","https://openalex.org/W2235056388","https://openalex.org/W2263557086","https://openalex.org/W2507742300","https://openalex.org/W2509359624","https://openalex.org/W2579889809","https://openalex.org/W2602608495","https://openalex.org/W2607456320","https://openalex.org/W2763081248","https://openalex.org/W2946222661","https://openalex.org/W2963136325","https://openalex.org/W2963433607","https://openalex.org/W2963492671","https://openalex.org/W2965558357","https://openalex.org/W2975608980","https://openalex.org/W4205654174","https://openalex.org/W4229650096","https://openalex.org/W4230058301","https://openalex.org/W4246367501","https://openalex.org/W6602268946","https://openalex.org/W6607309228","https://openalex.org/W6684701029"],"related_works":["https://openalex.org/W2155019053","https://openalex.org/W2044351385","https://openalex.org/W4226238329","https://openalex.org/W2074655359","https://openalex.org/W2369675785","https://openalex.org/W2042246561","https://openalex.org/W2048876835","https://openalex.org/W2087062690","https://openalex.org/W2587702371","https://openalex.org/W3120044416"],"abstract_inverted_index":{"Abstract":[0],"For":[1],"finite-dimensional":[2],"problems,":[3,66],"stochastic":[4,13,35,60,122],"approximation":[5],"methods":[6],"have":[7,84],"long":[8],"been":[9],"used":[10],"to":[11,18,40,83,116,152],"solve":[12],"optimization":[14,45,90],"problems.":[15],"Their":[16],"application":[17],"infinite-dimensional":[19],"problems":[20,46],"is":[21,71,77,81,92],"less":[22],"understood,":[23],"particularly":[24],"for":[25,33,62,157],"nonconvex":[26,63,75],"objectives.":[27],"This":[28],"paper":[29],"presents":[30],"convergence":[31,103,159],"results":[32],"the":[34,68,74,78,109,114,121,161],"proximal":[36,123],"gradient":[37,124],"method":[38],"applied":[39],"Hilbert":[41,97],"spaces,":[42],"motivated":[43],"by":[44,113,138],"with":[47,53,130],"partial":[48],"differential":[49],"equation":[50],"(PDE)":[51],"constraints":[52],"random":[54,110],"inputs":[55],"and":[56,64,73,142,148,163],"coefficients.":[57],"We":[58,99,119,154],"study":[59],"algorithms":[61],"nonsmooth":[65,69],"where":[67,145],"part":[70,76],"convex":[72],"expectation,":[79],"which":[80],"assumed":[82],"a":[85,96,127,131,139,165],"Lipschitz":[86],"continuous":[87],"gradient.":[88],"The":[89],"variable":[91],"an":[93],"element":[94],"of":[95,104,108,160],"space.":[98],"show":[100,164],"almost":[101],"sure":[102],"strong":[105],"limit":[106],"points":[107],"sequence":[111],"generated":[112],"algorithm":[115,125,162],"stationary":[117],"points.":[118],"demonstrate":[120],"on":[126],"tracking-type":[128],"functional":[129],"$$L^1$$":[132],"<mml:math":[133],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msup><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msup></mml:math>":[134],"-penalty":[135],"term":[136],"constrained":[137],"semilinear":[140],"PDE":[141],"box":[143],"constraints,":[144],"input":[146],"terms":[147],"coefficients":[149],"are":[150],"subject":[151],"uncertainty.":[153],"verify":[155],"conditions":[156],"ensuring":[158],"simulation.":[166]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":3},{"year":2023,"cited_by_count":6},{"year":2022,"cited_by_count":4},{"year":2021,"cited_by_count":2}],"updated_date":"2026-08-06T08:24:18.245995","created_date":"2020-01-10T00:00:00"}