{"id":"https://openalex.org/W1991607307","doi":"https://doi.org/10.1137/0806015","title":"A Unified Analysis of Hoffman\u2019s Bound via Fenchel Duality","display_name":"A Unified Analysis of Hoffman\u2019s Bound via Fenchel Duality","publication_year":1996,"publication_date":"1996-05-01","ids":{"openalex":"https://openalex.org/W1991607307","doi":"https://doi.org/10.1137/0806015","mag":"1991607307"},"language":"en","primary_location":{"id":"doi:10.1137/0806015","is_oa":false,"landing_page_url":"https://doi.org/10.1137/0806015","pdf_url":null,"source":{"id":"https://openalex.org/S928796702","display_name":"SIAM Journal on Optimization","issn_l":"1052-6234","issn":["1052-6234","1095-7189"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Optimization","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/A5085980354","display_name":"James V. Burke","orcid":"https://orcid.org/0000-0002-0215-7306"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"James V. Burke","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5110258450","display_name":"Paul Tseng","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Paul Tseng","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":6.7426,"has_fulltext":false,"cited_by_count":63,"citation_normalized_percentile":{"value":0.97255261,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"6","issue":"2","first_page":"265","last_page":"282"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.9995999932289124,"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":0.9995999932289124,"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":0.9979000091552734,"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/T10792","display_name":"Matrix Theory and Algorithms","score":0.9970999956130981,"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/mathematics","display_name":"Mathematics","score":0.8414837718009949},{"id":"https://openalex.org/keywords/bounded-function","display_name":"Bounded function","score":0.683593213558197},{"id":"https://openalex.org/keywords/convex-cone","display_name":"Convex cone","score":0.631881833076477},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.5853150486946106},{"id":"https://openalex.org/keywords/convex-set","display_name":"Convex set","score":0.5325597524642944},{"id":"https://openalex.org/keywords/infimum-and-supremum","display_name":"Infimum and supremum","score":0.5287194848060608},{"id":"https://openalex.org/keywords/norm","display_name":"Norm (philosophy)","score":0.5160183906555176},{"id":"https://openalex.org/keywords/subderivative","display_name":"Subderivative","score":0.5095984935760498},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.507946789264679},{"id":"https://openalex.org/keywords/duality","display_name":"Duality (order theory)","score":0.4960664212703705},{"id":"https://openalex.org/keywords/normed-vector-space","display_name":"Normed vector space","score":0.48355886340141296},{"id":"https://openalex.org/keywords/regular-polygon","display_name":"Regular polygon","score":0.4743023216724396},{"id":"https://openalex.org/keywords/convex-analysis","display_name":"Convex analysis","score":0.4555160105228424},{"id":"https://openalex.org/keywords/convex-body","display_name":"Convex body","score":0.4498714804649353},{"id":"https://openalex.org/keywords/operator-norm","display_name":"Operator norm","score":0.4462260603904724},{"id":"https://openalex.org/keywords/bounded-operator","display_name":"Bounded operator","score":0.4179125726222992},{"id":"https://openalex.org/keywords/convex-hull","display_name":"Convex hull","score":0.4142424464225769},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.2935899794101715},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.283719003200531},{"id":"https://openalex.org/keywords/convex-optimization","display_name":"Convex optimization","score":0.25385165214538574},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.13846167922019958}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8414837718009949},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.683593213558197},{"id":"https://openalex.org/C28340159","wikidata":"https://www.wikidata.org/wiki/Q2256541","display_name":"Convex cone","level":5,"score":0.631881833076477},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5853150486946106},{"id":"https://openalex.org/C49870271","wikidata":"https://www.wikidata.org/wiki/Q193657","display_name":"Convex set","level":4,"score":0.5325597524642944},{"id":"https://openalex.org/C95611797","wikidata":"https://www.wikidata.org/wiki/Q17502105","display_name":"Infimum and supremum","level":2,"score":0.5287194848060608},{"id":"https://openalex.org/C191795146","wikidata":"https://www.wikidata.org/wiki/Q3878446","display_name":"Norm (philosophy)","level":2,"score":0.5160183906555176},{"id":"https://openalex.org/C200661725","wikidata":"https://www.wikidata.org/wiki/Q289500","display_name":"Subderivative","level":4,"score":0.5095984935760498},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.507946789264679},{"id":"https://openalex.org/C2778023678","wikidata":"https://www.wikidata.org/wiki/Q554403","display_name":"Duality (order theory)","level":2,"score":0.4960664212703705},{"id":"https://openalex.org/C171739935","wikidata":"https://www.wikidata.org/wiki/Q726210","display_name":"Normed vector space","level":2,"score":0.48355886340141296},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.4743023216724396},{"id":"https://openalex.org/C12108790","wikidata":"https://www.wikidata.org/wiki/Q2234833","display_name":"Convex analysis","level":4,"score":0.4555160105228424},{"id":"https://openalex.org/C134912446","wikidata":"https://www.wikidata.org/wiki/Q5166516","display_name":"Convex body","level":4,"score":0.4498714804649353},{"id":"https://openalex.org/C68386474","wikidata":"https://www.wikidata.org/wiki/Q980557","display_name":"Operator norm","level":3,"score":0.4462260603904724},{"id":"https://openalex.org/C175454919","wikidata":"https://www.wikidata.org/wiki/Q2342396","display_name":"Bounded operator","level":3,"score":0.4179125726222992},{"id":"https://openalex.org/C206194317","wikidata":"https://www.wikidata.org/wiki/Q1138624","display_name":"Convex hull","level":3,"score":0.4142424464225769},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.2935899794101715},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.283719003200531},{"id":"https://openalex.org/C157972887","wikidata":"https://www.wikidata.org/wiki/Q463359","display_name":"Convex optimization","level":3,"score":0.25385165214538574},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.13846167922019958},{"id":"https://openalex.org/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","level":1,"score":0.0},{"id":"https://openalex.org/C17744445","wikidata":"https://www.wikidata.org/wiki/Q36442","display_name":"Political science","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/0806015","is_oa":false,"landing_page_url":"https://doi.org/10.1137/0806015","pdf_url":null,"source":{"id":"https://openalex.org/S928796702","display_name":"SIAM Journal on Optimization","issn_l":"1052-6234","issn":["1052-6234","1095-7189"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Optimization","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":35,"referenced_works":["https://openalex.org/W1534416612","https://openalex.org/W1632601927","https://openalex.org/W1964879221","https://openalex.org/W1964995199","https://openalex.org/W1966394100","https://openalex.org/W1978320424","https://openalex.org/W1990883342","https://openalex.org/W1992769584","https://openalex.org/W2012423427","https://openalex.org/W2019725228","https://openalex.org/W2020061185","https://openalex.org/W2020084878","https://openalex.org/W2021897193","https://openalex.org/W2033604629","https://openalex.org/W2042614948","https://openalex.org/W2045979166","https://openalex.org/W2047341913","https://openalex.org/W2066179689","https://openalex.org/W2071402680","https://openalex.org/W2074857865","https://openalex.org/W2075763272","https://openalex.org/W2085154531","https://openalex.org/W2085563112","https://openalex.org/W2090084467","https://openalex.org/W2095929557","https://openalex.org/W2112914259","https://openalex.org/W2115410170","https://openalex.org/W2145778743","https://openalex.org/W2170166494","https://openalex.org/W2334285086","https://openalex.org/W3210839039","https://openalex.org/W4241141003","https://openalex.org/W4241490660","https://openalex.org/W4249667877","https://openalex.org/W4293820388"],"related_works":["https://openalex.org/W3123914979","https://openalex.org/W2794086404","https://openalex.org/W3011352665","https://openalex.org/W4300098584","https://openalex.org/W2789172848","https://openalex.org/W2035999820","https://openalex.org/W2118722273","https://openalex.org/W2810523487","https://openalex.org/W3002864397","https://openalex.org/W4298313109"],"abstract_inverted_index":{"In":[0,43],"1952,":[1],"Hoffman":[2],"showed":[3],"that":[4],"the":[5,11,25,28,60,63,74,84,110,129,147,154,157,169,180],"distance":[6,64,155],"from":[7,98],"any":[8],"point":[9],"to":[10,53,128,152,179],"solution":[12],"set":[13],"of":[14,27,50,83,109,113,149,159,168,171,174],"a":[15,22,54,68,80,88,94,118,160,165,185,192],"linear":[16,75,96,101],"system":[17,76],"is":[18,65,77,93,106,117,144,184],"bounded":[19,166],"above":[20],"by":[21,67,79],"constant":[23],"times":[24],"norm":[26,61],"residual.":[29],"Subsequently,":[30],"this":[31,44],"bound":[32,52],"has":[33,38],"been":[34],"studied":[35],"extensively":[36],"and":[37,73,115,134],"found":[39],"many":[40,136],"practical":[41],"uses.":[42],"paper,":[45],"we":[46,132],"consider":[47],"an":[48,107],"extension":[49],"Hoffman\u2019s":[51,140],"partially":[55],"infinite-dimensional":[56],"setting":[57],"in":[58,123],"which":[59],"defining":[62],"replaced":[66,78],"positively":[69],"homogeneous":[70],"convex":[71,81,121,188,193],"function":[72,163],"inclusion":[82],"form":[85],"$Ax":[86],"-":[87],"\\in":[89],"K$,":[90],"where":[91,182],"A":[92],"continuous":[95],"operator":[97],"some":[99],"normed":[100],"space":[102,112],"into":[103],"$\\mathbf{R}^m":[104,124],",a$":[105],"element":[108],"range":[111],"A,":[114],"K":[116,183],"nonempty":[119,186],"closed":[120,187],"cone":[122],"$.":[125],"When":[126],"specialized":[127],"finite-dimensional":[130],"case,":[131],"unify":[133],"extend":[135],"existing":[137],"results":[138],"on":[139,146],"bound.":[141],"Our":[142],"analysis":[143,176],"based":[145],"use":[148],"Fenchel":[150],"duality":[151],"express":[153],"as":[156],"supremum":[158],"certain":[161],"concave":[162],"over":[164],"subset":[167],"polar":[170],"K.":[172],"Much":[173],"our":[175],"also":[177],"extends":[178],"case":[181],"set,":[189],"not":[190],"only":[191],"cone.":[194]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":3},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":1},{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":1},{"year":2012,"cited_by_count":7}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}
