{"id":"https://openalex.org/W2743133559","doi":"https://doi.org/10.1137/16m1084754","title":"Difference-of-Convex Learning: Directional Stationarity, Optimality, and Sparsity","display_name":"Difference-of-Convex Learning: Directional Stationarity, Optimality, and Sparsity","publication_year":2017,"publication_date":"2017-01-01","ids":{"openalex":"https://openalex.org/W2743133559","doi":"https://doi.org/10.1137/16m1084754","mag":"2743133559"},"language":"en","primary_location":{"id":"doi:10.1137/16m1084754","is_oa":false,"landing_page_url":"https://doi.org/10.1137/16m1084754","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/A5014074840","display_name":"Miju Ahn","orcid":"https://orcid.org/0000-0002-5070-4079"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Miju Ahn","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5002789695","display_name":"Jong\u2010Shi Pang","orcid":"https://orcid.org/0000-0002-2224-6787"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Jong-Shi Pang","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5059708262","display_name":"Jack Xin","orcid":"https://orcid.org/0000-0002-6438-8476"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Jack Xin","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5014074840"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":9.136,"has_fulltext":false,"cited_by_count":98,"citation_normalized_percentile":{"value":0.98897433,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":94,"max":100},"biblio":{"volume":"27","issue":"3","first_page":"1637","last_page":"1665"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.9994000196456909,"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":0.9994000196456909,"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/T10136","display_name":"Statistical Methods and Inference","score":0.9825999736785889,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"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/T12879","display_name":"Distributed Sensor Networks and Detection Algorithms","score":0.9294999837875366,"subfield":{"id":"https://openalex.org/subfields/1705","display_name":"Computer Networks and Communications"},"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/mathematical-optimization","display_name":"Mathematical optimization","score":0.6834617853164673},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6690526008605957},{"id":"https://openalex.org/keywords/penalty-method","display_name":"Penalty method","score":0.6444499492645264},{"id":"https://openalex.org/keywords/stationary-point","display_name":"Stationary point","score":0.6348011493682861},{"id":"https://openalex.org/keywords/augmented-lagrangian-method","display_name":"Augmented Lagrangian method","score":0.6334179639816284},{"id":"https://openalex.org/keywords/convex-optimization","display_name":"Convex optimization","score":0.4764569103717804},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.4729337990283966},{"id":"https://openalex.org/keywords/regularization","display_name":"Regularization (linguistics)","score":0.46466338634490967},{"id":"https://openalex.org/keywords/regular-polygon","display_name":"Regular polygon","score":0.4468477964401245},{"id":"https://openalex.org/keywords/optimization-problem","display_name":"Optimization problem","score":0.43731889128685},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.3446553647518158},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.2630664110183716},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.10396918654441833}],"concepts":[{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.6834617853164673},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6690526008605957},{"id":"https://openalex.org/C6180225","wikidata":"https://www.wikidata.org/wiki/Q3411771","display_name":"Penalty method","level":2,"score":0.6444499492645264},{"id":"https://openalex.org/C189237950","wikidata":"https://www.wikidata.org/wiki/Q2500758","display_name":"Stationary point","level":2,"score":0.6348011493682861},{"id":"https://openalex.org/C150452318","wikidata":"https://www.wikidata.org/wiki/Q4820432","display_name":"Augmented Lagrangian method","level":2,"score":0.6334179639816284},{"id":"https://openalex.org/C157972887","wikidata":"https://www.wikidata.org/wiki/Q463359","display_name":"Convex optimization","level":3,"score":0.4764569103717804},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.4729337990283966},{"id":"https://openalex.org/C2776135515","wikidata":"https://www.wikidata.org/wiki/Q17143721","display_name":"Regularization (linguistics)","level":2,"score":0.46466338634490967},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.4468477964401245},{"id":"https://openalex.org/C137836250","wikidata":"https://www.wikidata.org/wiki/Q984063","display_name":"Optimization problem","level":2,"score":0.43731889128685},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3446553647518158},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.2630664110183716},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.10396918654441833},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.0},{"id":"https://openalex.org/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","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}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/16m1084754","is_oa":false,"landing_page_url":"https://doi.org/10.1137/16m1084754","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":[{"id":"https://openalex.org/G1700293460","display_name":null,"funder_award_id":"IIS-1632971","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G5479139866","display_name":null,"funder_award_id":"IIS-1632935","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G6310009209","display_name":null,"funder_award_id":"DMS-1222507","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G6373369451","display_name":null,"funder_award_id":"CMMI-1333902","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G8243182001","display_name":null,"funder_award_id":"FA9550-15-1-0126","funder_id":"https://openalex.org/F4320338279","funder_display_name":"Air Force Office of Scientific Research"},{"id":"https://openalex.org/G8390493754","display_name":null,"funder_award_id":"DMS-1522383","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320338279","display_name":"Air Force Office of Scientific Research","ror":"https://ror.org/011e9bt93"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":60,"referenced_works":["https://openalex.org/W1719995771","https://openalex.org/W1965125844","https://openalex.org/W1965169081","https://openalex.org/W1968694834","https://openalex.org/W1969752902","https://openalex.org/W1986931325","https://openalex.org/W1992276332","https://openalex.org/W1993266883","https://openalex.org/W1996287810","https://openalex.org/W1996352909","https://openalex.org/W2018778093","https://openalex.org/W2020925091","https://openalex.org/W2028781966","https://openalex.org/W2034260606","https://openalex.org/W2040296279","https://openalex.org/W2040708309","https://openalex.org/W2057652230","https://openalex.org/W2063978378","https://openalex.org/W2074682976","https://openalex.org/W2079775628","https://openalex.org/W2089106993","https://openalex.org/W2090963365","https://openalex.org/W2093347109","https://openalex.org/W2101083207","https://openalex.org/W2103215959","https://openalex.org/W2107861471","https://openalex.org/W2116581043","https://openalex.org/W2122825543","https://openalex.org/W2129131372","https://openalex.org/W2129554161","https://openalex.org/W2129638195","https://openalex.org/W2133593515","https://openalex.org/W2135046866","https://openalex.org/W2146944586","https://openalex.org/W2149144946","https://openalex.org/W2150940164","https://openalex.org/W2154332973","https://openalex.org/W2166956514","https://openalex.org/W2225235077","https://openalex.org/W2296616510","https://openalex.org/W2340859457","https://openalex.org/W2471604734","https://openalex.org/W2507840378","https://openalex.org/W2738241071","https://openalex.org/W2809317144","https://openalex.org/W2950933264","https://openalex.org/W2963769561","https://openalex.org/W2964325628","https://openalex.org/W2964346891","https://openalex.org/W3081544918","https://openalex.org/W3098306969","https://openalex.org/W3099718522","https://openalex.org/W3099897794","https://openalex.org/W3100817920","https://openalex.org/W3103298265","https://openalex.org/W3104950855","https://openalex.org/W3105034597","https://openalex.org/W3106108064","https://openalex.org/W3122833167","https://openalex.org/W4250955649"],"related_works":["https://openalex.org/W4313189992","https://openalex.org/W2615021531","https://openalex.org/W1903454366","https://openalex.org/W1788398095","https://openalex.org/W2076134677","https://openalex.org/W1989415423","https://openalex.org/W1948655259","https://openalex.org/W3105202832","https://openalex.org/W59247221","https://openalex.org/W2478018497"],"abstract_inverted_index":{"This":[0],"paper":[1,80],"studies":[2],"a":[3,18,22,48,74,93,104],"fundamental":[4,83],"bicriteria":[5,99],"optimization":[6,100,126],"problem":[7,127,132,155],"for":[8,210,236],"variable":[9],"selection":[10],"in":[11,119,133,142],"statistical":[12,242],"learning;":[13],"the":[14,33,36,42,51,54,59,62,69,98,120,124,129,150,154,171,175,189,207,211,215,221,241],"two":[15],"criteria":[16],"are":[17,157,179,229],"loss/residual":[19],"function":[20,31,44,64,72],"and":[21,41,68,85,195,225,233],"model":[23,38,70],"control":[24,49,71],"(also":[25],"called":[26],"regularization,":[27],"penalty).":[28],"The":[29,199],"former":[30],"measures":[32],"fitness":[34],"of":[35,50,53,88,97,109,135,153,174,220,240],"learning":[37,243],"to":[39,92,144,149,161,185,204,214,223],"data":[40],"latter":[43],"is":[45,65,73,141],"employed":[46],"as":[47],"complexity":[52],"model.":[55],"We":[56,122],"focus":[57],"on":[58,103],"case":[60],"where":[61],"loss":[63],"(strongly)":[66],"convex":[67],"difference-of-convex":[75],"(dc)":[76],"sparsity":[77,86,112,227,234],"measure.":[78],"Our":[79],"establishes":[81],"some":[82],"optimality":[84,232],"properties":[87],"directional":[89],"stationary":[90],"solutions":[91,173],"nonconvex":[94,176,238],"Lagrangian":[95,125,177,212],"formulation":[96,178,213],"problem,":[101],"based":[102],"specially":[105],"structured":[106],"dc":[107],"representation":[108],"many":[110],"well-known":[111],"functions":[113,228],"that":[114,147],"can":[115],"be":[116],"profitably":[117],"exploited":[118],"analysis.":[121],"relate":[123],"with":[128],"penalty":[130,216],"constrained":[131,217],"terms":[134],"their":[136],"respective":[137],"d(irectional)-stationary":[138,172],"solutions;":[139],"this":[140],"contrast":[143],"common":[145],"analysis":[146,194],"pertains":[148],"(global)":[151],"minimizers":[152,181],"which":[156,170],"not":[158],"computable":[159],"due":[160,184],"nonconvexity.":[162],"Most":[163],"importantly,":[164],"we":[165],"provide":[166],"sufficient":[167],"conditions":[168,222],"under":[169],"global":[180],"(possibly":[182],"restricted":[183],"nondifferentiability),":[186],"thereby":[187],"filling":[188],"gap":[190],"between":[191],"previous":[192],"minimizer-based":[193],"practical":[196],"computational":[197],"considerations.":[198],"established":[200],"relation":[201],"allows":[202],"us":[203],"readily":[205],"apply":[206],"derived":[208],"results":[209,235],"formulation.":[218],"Specializations":[219],"exact":[224],"surrogate":[226],"discussed,":[230],"yielding":[231],"existing":[237],"formulations":[239],"problem.":[244]},"counts_by_year":[{"year":2026,"cited_by_count":2},{"year":2025,"cited_by_count":6},{"year":2024,"cited_by_count":6},{"year":2023,"cited_by_count":12},{"year":2022,"cited_by_count":17},{"year":2021,"cited_by_count":18},{"year":2020,"cited_by_count":8},{"year":2019,"cited_by_count":16},{"year":2018,"cited_by_count":11},{"year":2017,"cited_by_count":2}],"updated_date":"2026-04-10T15:06:20.359241","created_date":"2025-10-10T00:00:00"}
