{"id":"https://openalex.org/W4321769656","doi":"https://doi.org/10.1007/s11590-023-01981-2","title":"Conditions for linear convergence of the gradient method for non-convex optimization","display_name":"Conditions for linear convergence of the gradient method for non-convex optimization","publication_year":2023,"publication_date":"2023-02-25","ids":{"openalex":"https://openalex.org/W4321769656","doi":"https://doi.org/10.1007/s11590-023-01981-2"},"language":"en","primary_location":{"id":"doi:10.1007/s11590-023-01981-2","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s11590-023-01981-2","pdf_url":"https://link.springer.com/content/pdf/10.1007/s11590-023-01981-2.pdf","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://link.springer.com/content/pdf/10.1007/s11590-023-01981-2.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5030528113","display_name":"Hadi Abbaszadehpeivasti","orcid":"https://orcid.org/0000-0003-4652-8823"},"institutions":[{"id":"https://openalex.org/I193700539","display_name":"Tilburg University","ror":"https://ror.org/04b8v1s79","country_code":"NL","type":"education","lineage":["https://openalex.org/I193700539"]}],"countries":["NL"],"is_corresponding":true,"raw_author_name":"Hadi Abbaszadehpeivasti","raw_affiliation_strings":["Department of Econometrics and Operations Research, Tilburg University, Tilburg, The Netherlands"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Econometrics and Operations Research, Tilburg University, Tilburg, The Netherlands","institution_ids":["https://openalex.org/I193700539"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5019341464","display_name":"Etienne de Klerk","orcid":"https://orcid.org/0000-0003-3377-0063"},"institutions":[{"id":"https://openalex.org/I193700539","display_name":"Tilburg University","ror":"https://ror.org/04b8v1s79","country_code":"NL","type":"education","lineage":["https://openalex.org/I193700539"]}],"countries":["NL"],"is_corresponding":false,"raw_author_name":"Etienne de Klerk","raw_affiliation_strings":["Department of Econometrics and Operations Research, Tilburg University, Tilburg, The Netherlands"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Econometrics and Operations Research, Tilburg University, Tilburg, The Netherlands","institution_ids":["https://openalex.org/I193700539"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5011732780","display_name":"Moslem Zamani","orcid":"https://orcid.org/0000-0003-4086-0999"},"institutions":[{"id":"https://openalex.org/I193700539","display_name":"Tilburg University","ror":"https://ror.org/04b8v1s79","country_code":"NL","type":"education","lineage":["https://openalex.org/I193700539"]}],"countries":["NL"],"is_corresponding":false,"raw_author_name":"Moslem Zamani","raw_affiliation_strings":["Department of Econometrics and Operations Research, Tilburg University, Tilburg, The Netherlands"],"raw_orcid":"https://orcid.org/0000-0003-4086-0999","affiliations":[{"raw_affiliation_string":"Department of Econometrics and Operations Research, Tilburg University, Tilburg, The Netherlands","institution_ids":["https://openalex.org/I193700539"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5030528113"],"corresponding_institution_ids":["https://openalex.org/I193700539"],"apc_list":{"value":2290,"currency":"EUR","value_usd":2890},"apc_paid":{"value":2290,"currency":"EUR","value_usd":2890},"fwci":1.2423,"has_fulltext":true,"cited_by_count":6,"citation_normalized_percentile":{"value":0.73591549,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":91,"max":97},"biblio":{"volume":"17","issue":"5","first_page":"1105","last_page":"1125"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.9980000257492065,"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":0.9980000257492065,"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/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.9955000281333923,"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/T11205","display_name":"Numerical methods in inverse problems","score":0.9925000071525574,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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":0.7684127688407898},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.6746322512626648},{"id":"https://openalex.org/keywords/rate-of-convergence","display_name":"Rate of convergence","score":0.6381438374519348},{"id":"https://openalex.org/keywords/proximal-gradient-methods","display_name":"Proximal Gradient Methods","score":0.59635990858078},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5237548351287842},{"id":"https://openalex.org/keywords/convex-function","display_name":"Convex function","score":0.4976079761981964},{"id":"https://openalex.org/keywords/regular-polygon","display_name":"Regular polygon","score":0.48283180594444275},{"id":"https://openalex.org/keywords/gradient-method","display_name":"Gradient method","score":0.48059412837028503},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.4757423400878906},{"id":"https://openalex.org/keywords/computational-intelligence","display_name":"Computational intelligence","score":0.4658670425415039},{"id":"https://openalex.org/keywords/convex-optimization","display_name":"Convex optimization","score":0.44843217730522156},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.15366312861442566},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.056262433528900146}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7684127688407898},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.6746322512626648},{"id":"https://openalex.org/C57869625","wikidata":"https://www.wikidata.org/wiki/Q1783502","display_name":"Rate of convergence","level":3,"score":0.6381438374519348},{"id":"https://openalex.org/C10494615","wikidata":"https://www.wikidata.org/wiki/Q17086765","display_name":"Proximal Gradient Methods","level":4,"score":0.59635990858078},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5237548351287842},{"id":"https://openalex.org/C145446738","wikidata":"https://www.wikidata.org/wiki/Q319913","display_name":"Convex function","level":3,"score":0.4976079761981964},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.48283180594444275},{"id":"https://openalex.org/C115680565","wikidata":"https://www.wikidata.org/wiki/Q5977448","display_name":"Gradient method","level":2,"score":0.48059412837028503},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.4757423400878906},{"id":"https://openalex.org/C139502532","wikidata":"https://www.wikidata.org/wiki/Q1122090","display_name":"Computational intelligence","level":2,"score":0.4658670425415039},{"id":"https://openalex.org/C157972887","wikidata":"https://www.wikidata.org/wiki/Q463359","display_name":"Convex optimization","level":3,"score":0.44843217730522156},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.15366312861442566},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.056262433528900146},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"score":0.0},{"id":"https://openalex.org/C50522688","wikidata":"https://www.wikidata.org/wiki/Q189833","display_name":"Economic growth","level":1,"score":0.0},{"id":"https://openalex.org/C31258907","wikidata":"https://www.wikidata.org/wiki/Q1301371","display_name":"Computer network","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},{"id":"https://openalex.org/C127162648","wikidata":"https://www.wikidata.org/wiki/Q16858953","display_name":"Channel (broadcasting)","level":2,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1007/s11590-023-01981-2","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s11590-023-01981-2","pdf_url":"https://link.springer.com/content/pdf/10.1007/s11590-023-01981-2.pdf","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:tilburguniversity.edu:openaire/a5bebd71-1bf3-48ee-a3b9-fec3d19d9a02","is_oa":true,"landing_page_url":"https://research.tilburguniversity.edu/en/publications/a5bebd71-1bf3-48ee-a3b9-fec3d19d9a02","pdf_url":null,"source":{"id":"https://openalex.org/S4406923027","display_name":"Tilburg University Research Portal","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":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Abbaszadehpeivasti, H, de Klerk, E & Zamani, M 2023, 'Conditions for linear convergence of the gradient method for non-convex optimization', Optimization Letters, vol. 17, pp. 1105\u20131125. https://doi.org/10.1007/s11590-023-01981-2","raw_type":"info:eu-repo/semantics/publishedVersion"}],"best_oa_location":{"id":"doi:10.1007/s11590-023-01981-2","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s11590-023-01981-2","pdf_url":"https://link.springer.com/content/pdf/10.1007/s11590-023-01981-2.pdf","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":[{"display_name":"Reduced inequalities","id":"https://metadata.un.org/sdg/10","score":0.6499999761581421}],"awards":[{"id":"https://openalex.org/G845402991","display_name":null,"funder_award_id":"2019.015","funder_id":"https://openalex.org/F4320321800","funder_display_name":"Nederlandse Organisatie voor Wetenschappelijk Onderzoek"}],"funders":[{"id":"https://openalex.org/F4320321800","display_name":"Nederlandse Organisatie voor Wetenschappelijk Onderzoek","ror":"https://ror.org/04jsz6e67"}],"has_content":{"pdf":true,"grobid_xml":false},"content_urls":{"pdf":"https://content.openalex.org/works/W4321769656.pdf"},"referenced_works_count":17,"referenced_works":["https://openalex.org/W1962121538","https://openalex.org/W1979896658","https://openalex.org/W1981042292","https://openalex.org/W2030161963","https://openalex.org/W2078853581","https://openalex.org/W2129732816","https://openalex.org/W2473376220","https://openalex.org/W2901838001","https://openalex.org/W2963190258","https://openalex.org/W2963248893","https://openalex.org/W2963830980","https://openalex.org/W3013479538","https://openalex.org/W3037153379","https://openalex.org/W3204743358","https://openalex.org/W6600599346","https://openalex.org/W6601403641","https://openalex.org/W6601968593"],"related_works":["https://openalex.org/W2963086517","https://openalex.org/W2564735875","https://openalex.org/W3123504125","https://openalex.org/W3147739796","https://openalex.org/W1983212821","https://openalex.org/W3127841625","https://openalex.org/W2594900731","https://openalex.org/W2811217697","https://openalex.org/W2364728921","https://openalex.org/W2348618647"],"abstract_inverted_index":{"Abstract":[0],"In":[1],"this":[2,49],"paper,":[3],"we":[4,71],"derive":[5],"a":[6,36],"new":[7],"linear":[8,42,67],"convergence":[9,43,68],"rate":[10],"for":[11,19,41,48,60],"the":[12,25,32,45,62,76],"gradient":[13,63],"method":[14,64],"with":[15,75],"fixed":[16],"step":[17],"lengths":[18],"non-convex":[20],"smooth":[21],"optimization":[22],"problems":[23],"satisfying":[24],"Polyak-\u0141ojasiewicz":[26],"(P\u0141)":[27],"inequality.":[28,78],"We":[29,53],"establish":[30],"that":[31],"P\u0141":[33,77],"inequality":[34],"is":[35],"necessary":[37],"and":[38],"sufficient":[39],"condition":[40],"to":[44],"optimal":[46],"value":[47],"class":[50],"of":[51,58],"problems.":[52],"list":[54],"some":[55],"related":[56],"classes":[57],"functions":[59],"which":[61],"may":[65],"enjoy":[66],"rate.":[69],"Moreover,":[70],"investigate":[72],"their":[73],"relationship":[74]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":3},{"year":2023,"cited_by_count":2}],"updated_date":"2025-12-21T01:58:51.020947","created_date":"2025-10-10T00:00:00"}