{"id":"https://openalex.org/W2026867783","doi":"https://doi.org/10.1287/moor.24.1.219","title":"Solutions to Affine Generalized Equations Using Proximal Mappings","display_name":"Solutions to Affine Generalized Equations Using Proximal Mappings","publication_year":1999,"publication_date":"1999-02-01","ids":{"openalex":"https://openalex.org/W2026867783","doi":"https://doi.org/10.1287/moor.24.1.219","mag":"2026867783"},"language":"en","primary_location":{"id":"doi:10.1287/moor.24.1.219","is_oa":false,"landing_page_url":"https://doi.org/10.1287/moor.24.1.219","pdf_url":null,"source":{"id":"https://openalex.org/S55826652","display_name":"Mathematics of Operations Research","issn_l":"0364-765X","issn":["0364-765X","1526-5471"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315699","host_organization_name":"Institute for Operations Research and the Management Sciences","host_organization_lineage":["https://openalex.org/P4310315699"],"host_organization_lineage_names":["Institute for Operations Research and the Management Sciences"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Operations Research","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://minds.wisconsin.edu/bitstream/1793/64582/1/94-15.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5026811780","display_name":"Stephen C. Billups","orcid":"https://orcid.org/0000-0003-3627-0793"},"institutions":[{"id":"https://openalex.org/I921990950","display_name":"University of Colorado Denver","ror":"https://ror.org/02hh7en24","country_code":"US","type":"education","lineage":["https://openalex.org/I921990950"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Stephen C. Billups","raw_affiliation_strings":["Department of Mathematics, University of Colorado, Denver, Colorado 80217","Department of Mathematics, University of Colorado, Denver, Colorado 80217#TAB#"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Colorado, Denver, Colorado 80217","institution_ids":["https://openalex.org/I921990950"]},{"raw_affiliation_string":"Department of Mathematics, University of Colorado, Denver, Colorado 80217#TAB#","institution_ids":["https://openalex.org/I921990950"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5088455558","display_name":"Michael C. Ferris","orcid":"https://orcid.org/0000-0002-7652-6776"},"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":false,"raw_author_name":"Michael C. Ferris","raw_affiliation_strings":["Computer Sciences Department, University of Wisconsin, Madison, Wisconsin 53706","Computer Sciences Department, University of Wisconsin, Madison, Wisconsin, 53706"],"affiliations":[{"raw_affiliation_string":"Computer Sciences Department, University of Wisconsin, Madison, Wisconsin 53706","institution_ids":["https://openalex.org/I135310074"]},{"raw_affiliation_string":"Computer Sciences Department, University of Wisconsin, Madison, Wisconsin, 53706","institution_ids":["https://openalex.org/I135310074"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5026811780"],"corresponding_institution_ids":["https://openalex.org/I921990950"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":6,"citation_normalized_percentile":{"value":0.18067061,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"24","issue":"1","first_page":"219","last_page":"236"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.9984999895095825,"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":0.9984999895095825,"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":0.9965000152587891,"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":0.9905999898910522,"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":0.9171597957611084},{"id":"https://openalex.org/keywords/affine-transformation","display_name":"Affine transformation","score":0.7317972779273987},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5323294401168823},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.36028262972831726},{"id":"https://openalex.org/keywords/calculus","display_name":"Calculus (dental)","score":0.32935720682144165},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.301798552274704}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.9171597957611084},{"id":"https://openalex.org/C92757383","wikidata":"https://www.wikidata.org/wiki/Q382497","display_name":"Affine transformation","level":2,"score":0.7317972779273987},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5323294401168823},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.36028262972831726},{"id":"https://openalex.org/C2777686260","wikidata":"https://www.wikidata.org/wiki/Q144037","display_name":"Calculus (dental)","level":2,"score":0.32935720682144165},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.301798552274704},{"id":"https://openalex.org/C199343813","wikidata":"https://www.wikidata.org/wiki/Q12128","display_name":"Dentistry","level":1,"score":0.0},{"id":"https://openalex.org/C71924100","wikidata":"https://www.wikidata.org/wiki/Q11190","display_name":"Medicine","level":0,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1287/moor.24.1.219","is_oa":false,"landing_page_url":"https://doi.org/10.1287/moor.24.1.219","pdf_url":null,"source":{"id":"https://openalex.org/S55826652","display_name":"Mathematics of Operations Research","issn_l":"0364-765X","issn":["0364-765X","1526-5471"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315699","host_organization_name":"Institute for Operations Research and the Management Sciences","host_organization_lineage":["https://openalex.org/P4310315699"],"host_organization_lineage_names":["Institute for Operations Research and the Management Sciences"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Operations Research","raw_type":"journal-article"},{"id":"pmh:oai:minds.wisconsin.edu:1793/64582","is_oa":true,"landing_page_url":"http://digital.library.wisc.edu/1793/64582","pdf_url":"https://minds.wisconsin.edu/bitstream/1793/64582/1/94-15.pdf","source":{"id":"https://openalex.org/S4306401622","display_name":"Minds at UW (University of Wisconsin)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I135310074","host_organization_name":"University of Wisconsin\u2013Madison","host_organization_lineage":["https://openalex.org/I135310074"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Technical Report"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.50.5401","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.50.5401","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"ftp://ftp.cs.wisc.edu/math-prog/tech-reports/94-15.ps","raw_type":"text"},{"id":"pmh:oai:RePEc:inm:ormoor:v:24:y:1999:i:1:p:219-236","is_oa":false,"landing_page_url":"http://doi.org/10.1287/moor.24.1.219","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":"","raw_type":"article"}],"best_oa_location":{"id":"pmh:oai:minds.wisconsin.edu:1793/64582","is_oa":true,"landing_page_url":"http://digital.library.wisc.edu/1793/64582","pdf_url":"https://minds.wisconsin.edu/bitstream/1793/64582/1/94-15.pdf","source":{"id":"https://openalex.org/S4306401622","display_name":"Minds at UW (University of Wisconsin)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I135310074","host_organization_name":"University of Wisconsin\u2013Madison","host_organization_lineage":["https://openalex.org/I135310074"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Technical Report"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G3337293965","display_name":"PYI:  Development of Efficient Serial and Parallel          Algorithms Based on Mathematical Programming Theory","funder_award_id":"9157632","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G5219260752","display_name":null,"funder_award_id":"DE-FG","funder_id":"https://openalex.org/F4320306084","funder_display_name":"U.S. Department of Energy"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320306084","display_name":"U.S. Department of Energy","ror":"https://ror.org/01bj3aw27"}],"has_content":{"pdf":true,"grobid_xml":false},"content_urls":{"pdf":"https://content.openalex.org/works/W2026867783.pdf"},"referenced_works_count":29,"referenced_works":["https://openalex.org/W61778493","https://openalex.org/W1481113074","https://openalex.org/W1484024929","https://openalex.org/W1499918501","https://openalex.org/W1528330203","https://openalex.org/W1582414340","https://openalex.org/W1582617215","https://openalex.org/W1583089965","https://openalex.org/W1977904123","https://openalex.org/W1979240183","https://openalex.org/W2000151369","https://openalex.org/W2001024683","https://openalex.org/W2007753746","https://openalex.org/W2040732788","https://openalex.org/W2042261731","https://openalex.org/W2047090956","https://openalex.org/W2048492904","https://openalex.org/W2064028560","https://openalex.org/W2089649903","https://openalex.org/W2105207928","https://openalex.org/W2128234186","https://openalex.org/W2162916973","https://openalex.org/W2462426969","https://openalex.org/W2595644070","https://openalex.org/W2948315098","https://openalex.org/W4241481124","https://openalex.org/W4248797359","https://openalex.org/W4249667877","https://openalex.org/W4388295056"],"related_works":["https://openalex.org/W2038416447","https://openalex.org/W2364151838","https://openalex.org/W2066926363","https://openalex.org/W2354159744","https://openalex.org/W2351539857","https://openalex.org/W2381972071","https://openalex.org/W4301382669","https://openalex.org/W1973641256","https://openalex.org/W2151040698","https://openalex.org/W4287759382"],"abstract_inverted_index":{"The":[0,122],"normal":[1,41,58],"map":[2],"has":[3],"proven":[4],"to":[5,43,60,107,135],"be":[6],"a":[7,32,54,81,87,139,142],"powerful":[8],"tool":[9],"for":[10,112],"solving":[11],"generalized":[12,110],"equations":[13,62],"of":[14,56,63,93,133,141],"the":[15,40,52,57,64,105,109,129,136],"form:":[16,67],"find":[17,68],"z":[18,69],"\u2208":[19,23,70,74],"C,":[20],"with":[21,72,104],"0":[22,73],"F(z)":[24,75],"+":[25,76],"N":[26,36],"c":[27,37],"(z),":[28],"where":[29,78,138],"C":[30,44],"is":[31,39,80,145],"convex":[33],"set":[34],"and":[35,98,118],"(z)":[38],"cone":[42],"at":[45],"z.":[46],"In":[47],"this":[48,101],"paper,":[49],"we":[50,99,125],"use":[51,100],"T-map,":[53,106],"generalization":[55],"map,":[59],"solve":[61,108],"more":[65],"general":[66],"dom(T),":[71],"T(z),":[77],"T":[79],"maximal":[82],"monotone":[83],"multifunction.":[84],"We":[85],"present":[86],"path-following":[88,123],"algorithm":[89,124],"that":[90],"determines":[91],"zeros":[92],"coherently":[94,114],"oriented":[95,115],"piecewise-affine":[96],"functions,":[97],"algorithm,":[102],"together":[103],"equation":[111],"affine,":[113],"functions":[116],"F,":[117],"polyhedral":[119],"multifunctions":[120],"T.":[121],"develop":[126],"here":[127],"extends":[128],"piecewise-linear":[130],"homotopy":[131],"framework":[132],"Eaves":[134],"case":[137],"representation":[140],"subdivided":[143],"manifold":[144],"unknown.":[146]},"counts_by_year":[{"year":2020,"cited_by_count":1}],"updated_date":"2026-04-21T08:09:41.155169","created_date":"2025-10-10T00:00:00"}