{"id":"https://openalex.org/W1980131437","doi":"https://doi.org/10.1137/08073682x","title":"Lipschitz Behavior of the Robust Regularization","display_name":"Lipschitz Behavior of the Robust Regularization","publication_year":2009,"publication_date":"2009-12-11","ids":{"openalex":"https://openalex.org/W1980131437","doi":"https://doi.org/10.1137/08073682x","mag":"1980131437"},"language":"en","primary_location":{"id":"doi:10.1137/08073682x","is_oa":false,"landing_page_url":"https://doi.org/10.1137/08073682x","pdf_url":null,"source":{"id":"https://openalex.org/S897311980","display_name":"SIAM Journal on Control and Optimization","issn_l":"0363-0129","issn":["0363-0129","1095-7138"],"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 Control and 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/A5081400874","display_name":"Adrian S. Lewis","orcid":"https://orcid.org/0000-0002-1785-1106"},"institutions":[{"id":"https://openalex.org/I205783295","display_name":"Cornell University","ror":"https://ror.org/05bnh6r87","country_code":"US","type":"education","lineage":["https://openalex.org/I205783295"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Adrian S. Lewis","raw_affiliation_strings":["aslewis@orie.cornell.edu#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"aslewis@orie.cornell.edu#TAB#","institution_ids":["https://openalex.org/I205783295"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5086990462","display_name":"C. H. Jeffrey Pang","orcid":"https://orcid.org/0000-0001-7209-3013"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"C. H. Jeffrey Pang","raw_affiliation_strings":["chj2pang@math.uwaterloo.ca#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"chj2pang@math.uwaterloo.ca#TAB#","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":4.2354,"has_fulltext":false,"cited_by_count":30,"citation_normalized_percentile":{"value":0.93703532,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"48","issue":"5","first_page":"3080","last_page":"3104"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.998199999332428,"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":0.998199999332428,"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":0.9973999857902527,"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/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.9926999807357788,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8579083681106567},{"id":"https://openalex.org/keywords/lipschitz-continuity","display_name":"Lipschitz continuity","score":0.7506594657897949},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.6100045442581177},{"id":"https://openalex.org/keywords/regularization","display_name":"Regularization (linguistics)","score":0.6082616448402405},{"id":"https://openalex.org/keywords/spectral-radius","display_name":"Spectral radius","score":0.5319759249687195},{"id":"https://openalex.org/keywords/algebraic-number","display_name":"Algebraic number","score":0.5021555423736572},{"id":"https://openalex.org/keywords/regular-polygon","display_name":"Regular polygon","score":0.4919709861278534},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4340224862098694},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.4077267050743103},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.3933231234550476},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.32173195481300354},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.08998715877532959},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.08101704716682434}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8579083681106567},{"id":"https://openalex.org/C22324862","wikidata":"https://www.wikidata.org/wiki/Q652707","display_name":"Lipschitz continuity","level":2,"score":0.7506594657897949},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.6100045442581177},{"id":"https://openalex.org/C2776135515","wikidata":"https://www.wikidata.org/wiki/Q17143721","display_name":"Regularization (linguistics)","level":2,"score":0.6082616448402405},{"id":"https://openalex.org/C140532419","wikidata":"https://www.wikidata.org/wiki/Q249748","display_name":"Spectral radius","level":3,"score":0.5319759249687195},{"id":"https://openalex.org/C9376300","wikidata":"https://www.wikidata.org/wiki/Q168817","display_name":"Algebraic number","level":2,"score":0.5021555423736572},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.4919709861278534},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4340224862098694},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4077267050743103},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3933231234550476},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.32173195481300354},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.08998715877532959},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.08101704716682434},{"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/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/08073682x","is_oa":false,"landing_page_url":"https://doi.org/10.1137/08073682x","pdf_url":null,"source":{"id":"https://openalex.org/S897311980","display_name":"SIAM Journal on Control and Optimization","issn_l":"0363-0129","issn":["0363-0129","1095-7138"],"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 Control and Optimization","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":28,"referenced_works":["https://openalex.org/W22403494","https://openalex.org/W26023427","https://openalex.org/W149522324","https://openalex.org/W175988912","https://openalex.org/W1485995001","https://openalex.org/W1495980450","https://openalex.org/W1551360398","https://openalex.org/W1568307856","https://openalex.org/W1875804059","https://openalex.org/W1960239420","https://openalex.org/W1996685707","https://openalex.org/W1998990712","https://openalex.org/W2002358177","https://openalex.org/W2015224767","https://openalex.org/W2020167116","https://openalex.org/W2033316062","https://openalex.org/W2045979166","https://openalex.org/W2047075189","https://openalex.org/W2049041383","https://openalex.org/W2081038760","https://openalex.org/W2081600571","https://openalex.org/W2100004457","https://openalex.org/W2130417105","https://openalex.org/W2143279738","https://openalex.org/W2153562582","https://openalex.org/W2162052349","https://openalex.org/W3140976322","https://openalex.org/W3210839039"],"related_works":["https://openalex.org/W3185235544","https://openalex.org/W2911623553","https://openalex.org/W4297791327","https://openalex.org/W2897842840","https://openalex.org/W2397777611","https://openalex.org/W1502031429","https://openalex.org/W2417585376","https://openalex.org/W1994157709","https://openalex.org/W4294336468","https://openalex.org/W2797811993"],"abstract_inverted_index":{"To":[0],"minimize":[1,13],"or":[2,14],"upper-bound":[3,15],"the":[4,16,21,27,31,52,70,85,91,100],"value":[5,29],"of":[6,30,55,65,99,111],"a":[7,24,56],"function":[8,32,87],"\u201crobustly,\u201d":[9],"we":[10,67],"might":[11],"instead":[12],"\u201c$\\epsilon$-robust":[17],"regularization,\u201d":[18],"defined":[19,113],"as":[20],"map":[22],"from":[23],"point":[25],"to":[26,41,46,59],"maximum":[28],"within":[33],"an":[34],"$\\epsilon$-radius.":[35],"This":[36],"regularization":[37,72],"may":[38],"be":[39],"easy":[40],"compute:":[42],"convex":[43],"quadratics":[44],"lead":[45],"semidefinite-representable":[47],"regularizations,":[48],"for":[49,79],"example,":[50],"and":[51,119],"spectral":[53,92],"radius":[54],"matrix":[57],"leads":[58],"pseudospectral":[60],"computations.":[61],"For":[62],"favorable":[63,96],"classes":[64],"functions,":[66],"show":[68],"that":[69,107],"robust":[71],"is":[73,88],"Lipschitz":[74],"around":[75],"any":[76],"given":[77],"point,":[78],"all":[80],"small":[81],"$\\epsilon>0$,":[82],"even":[83],"if":[84],"original":[86],"non-Lipschitz":[89],"(like":[90],"radius).":[93],"One":[94],"such":[95],"class":[97],"consists":[98],"semi-algebraic":[101],"functions.":[102],"Such":[103],"functions":[104],"have":[105],"graphs":[106],"are":[108,120],"finite":[109],"unions":[110],"sets":[112],"by":[114],"finitely":[115],"many":[116],"polynomial":[117],"inequalities,":[118],"commonly":[121],"encountered":[122],"in":[123],"applications.":[124]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":2},{"year":2021,"cited_by_count":2},{"year":2019,"cited_by_count":2},{"year":2018,"cited_by_count":4},{"year":2017,"cited_by_count":4},{"year":2014,"cited_by_count":1},{"year":2013,"cited_by_count":2},{"year":2012,"cited_by_count":8}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}
