{"id":"https://openalex.org/W1979459040","doi":"https://doi.org/10.1007/s101070050078","title":"Linear convergence of epsilon-subgradient descent methods for a class of convex functions","display_name":"Linear convergence of epsilon-subgradient descent methods for a class of convex functions","publication_year":1999,"publication_date":"1999-09-01","ids":{"openalex":"https://openalex.org/W1979459040","doi":"https://doi.org/10.1007/s101070050078","mag":"1979459040"},"language":"en","primary_location":{"id":"doi:10.1007/s101070050078","is_oa":false,"landing_page_url":"https://doi.org/10.1007/s101070050078","pdf_url":null,"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":null,"license_id":null,"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":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5005541868","display_name":"Stephen M. Robinson","orcid":"https://orcid.org/0000-0001-9845-2846"},"institutions":[{"id":"https://openalex.org/I135310074","display_name":"University of Wisconsin\u2013Madison","ror":"https://ror.org/01y2jtd41","country_code":"US","type":"education","lineage":["https://openalex.org/I135310074"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Stephen M. Robinson","raw_affiliation_strings":["Department of Industrial Engineering, University of Wisconsin\u2013Madison, 1513 University Avenue, Madison, WI 53706-1572, USA, e-mail: smrobins@facstaff.wisc.edu, , , , , , US","Department of Industrial Engineering, University of Wisconsin\u2013Madison, 1513 University Avenue, Madison, WI 53706-1572, USA, e-mail: smrobins@facstaff.wisc.edu, US"],"affiliations":[{"raw_affiliation_string":"Department of Industrial Engineering, University of Wisconsin\u2013Madison, 1513 University Avenue, Madison, WI 53706-1572, USA, e-mail: smrobins@facstaff.wisc.edu, , , , , , US","institution_ids":["https://openalex.org/I135310074"]},{"raw_affiliation_string":"Department of Industrial Engineering, University of Wisconsin\u2013Madison, 1513 University Avenue, Madison, WI 53706-1572, USA, e-mail: smrobins@facstaff.wisc.edu, US","institution_ids":["https://openalex.org/I135310074"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5005541868"],"corresponding_institution_ids":["https://openalex.org/I135310074"],"apc_list":{"value":2190,"currency":"EUR","value_usd":2890},"apc_paid":null,"fwci":0.5037,"has_fulltext":false,"cited_by_count":47,"citation_normalized_percentile":{"value":0.6310757,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"86","issue":"1","first_page":"41","last_page":"50"},"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/T12661","display_name":"Iterative Methods for Nonlinear Equations","score":0.9990000128746033,"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":0.9984999895095825,"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/subgradient-method","display_name":"Subgradient method","score":0.9379863739013672},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8731340169906616},{"id":"https://openalex.org/keywords/resolvent","display_name":"Resolvent","score":0.7010669112205505},{"id":"https://openalex.org/keywords/class","display_name":"Class (philosophy)","score":0.5711168646812439},{"id":"https://openalex.org/keywords/convex-function","display_name":"Convex function","score":0.5584290623664856},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.5250382423400879},{"id":"https://openalex.org/keywords/regular-polygon","display_name":"Regular polygon","score":0.5094233155250549},{"id":"https://openalex.org/keywords/descent","display_name":"Descent (aeronautics)","score":0.5066288113594055},{"id":"https://openalex.org/keywords/sequence","display_name":"Sequence (biology)","score":0.5022785663604736},{"id":"https://openalex.org/keywords/rate-of-convergence","display_name":"Rate of convergence","score":0.47633129358291626},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4433459937572479},{"id":"https://openalex.org/keywords/proximal-gradient-methods","display_name":"Proximal Gradient Methods","score":0.4425066411495209},{"id":"https://openalex.org/keywords/convex-optimization","display_name":"Convex optimization","score":0.4328819513320923},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.3469703197479248},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.33635085821151733},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.3122911751270294},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.08261778950691223},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.0769452154636383}],"concepts":[{"id":"https://openalex.org/C158968445","wikidata":"https://www.wikidata.org/wiki/Q7631150","display_name":"Subgradient method","level":2,"score":0.9379863739013672},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8731340169906616},{"id":"https://openalex.org/C196267783","wikidata":"https://www.wikidata.org/wiki/Q4392490","display_name":"Resolvent","level":2,"score":0.7010669112205505},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.5711168646812439},{"id":"https://openalex.org/C145446738","wikidata":"https://www.wikidata.org/wiki/Q319913","display_name":"Convex function","level":3,"score":0.5584290623664856},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.5250382423400879},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.5094233155250549},{"id":"https://openalex.org/C2776637919","wikidata":"https://www.wikidata.org/wiki/Q624380","display_name":"Descent (aeronautics)","level":2,"score":0.5066288113594055},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.5022785663604736},{"id":"https://openalex.org/C57869625","wikidata":"https://www.wikidata.org/wiki/Q1783502","display_name":"Rate of convergence","level":3,"score":0.47633129358291626},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4433459937572479},{"id":"https://openalex.org/C10494615","wikidata":"https://www.wikidata.org/wiki/Q17086765","display_name":"Proximal Gradient Methods","level":4,"score":0.4425066411495209},{"id":"https://openalex.org/C157972887","wikidata":"https://www.wikidata.org/wiki/Q463359","display_name":"Convex optimization","level":3,"score":0.4328819513320923},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3469703197479248},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.33635085821151733},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.3122911751270294},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.08261778950691223},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.0769452154636383},{"id":"https://openalex.org/C54355233","wikidata":"https://www.wikidata.org/wiki/Q7162","display_name":"Genetics","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/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"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/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","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},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","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/C146978453","wikidata":"https://www.wikidata.org/wiki/Q3798668","display_name":"Aerospace engineering","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1007/s101070050078","is_oa":false,"landing_page_url":"https://doi.org/10.1007/s101070050078","pdf_url":null,"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":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematical Programming","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[{"display_name":"Reduced inequalities","id":"https://metadata.un.org/sdg/10","score":0.4699999988079071}],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":13,"referenced_works":["https://openalex.org/W78423669","https://openalex.org/W1528330203","https://openalex.org/W1577338488","https://openalex.org/W2038497950","https://openalex.org/W2041928877","https://openalex.org/W2067243588","https://openalex.org/W2079688774","https://openalex.org/W2083432309","https://openalex.org/W2090963365","https://openalex.org/W2092921045","https://openalex.org/W2095030074","https://openalex.org/W2135779729","https://openalex.org/W2160068964"],"related_works":["https://openalex.org/W2905950575","https://openalex.org/W2073750241","https://openalex.org/W2465405765","https://openalex.org/W1863473538","https://openalex.org/W4319831228","https://openalex.org/W2804400349","https://openalex.org/W4297153560","https://openalex.org/W2964186284","https://openalex.org/W2735159666","https://openalex.org/W2962970637"],"abstract_inverted_index":null,"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":3},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":4},{"year":2019,"cited_by_count":3},{"year":2018,"cited_by_count":2},{"year":2017,"cited_by_count":3},{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":3},{"year":2013,"cited_by_count":1},{"year":2012,"cited_by_count":2}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}