{"id":"https://openalex.org/W4287268094","doi":"https://doi.org/10.1137/21m1442838","title":"Value-Gradient Based Formulation of Optimal Control Problem and Machine Learning Algorithm","display_name":"Value-Gradient Based Formulation of Optimal Control Problem and Machine Learning Algorithm","publication_year":2023,"publication_date":"2023-04-26","ids":{"openalex":"https://openalex.org/W4287268094","doi":"https://doi.org/10.1137/21m1442838"},"language":"en","primary_location":{"id":"doi:10.1137/21m1442838","is_oa":false,"landing_page_url":"https://doi.org/10.1137/21m1442838","pdf_url":null,"source":{"id":"https://openalex.org/S203348814","display_name":"SIAM Journal on Numerical Analysis","issn_l":"0036-1429","issn":["0036-1429","1095-7170"],"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 Numerical Analysis","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"bronze","oa_url":"https://scholars.cityu.edu.hk/en/publications/value-gradient-based-formulation-of-optimal-control-problem-and-m","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5027426452","display_name":"Alain Bensoussan","orcid":"https://orcid.org/0000-0003-2465-1249"},"institutions":[{"id":"https://openalex.org/I162577319","display_name":"The University of Texas at Dallas","ror":"https://ror.org/049emcs32","country_code":"US","type":"education","lineage":["https://openalex.org/I162577319"]},{"id":"https://openalex.org/I168719708","display_name":"City University of Hong Kong","ror":"https://ror.org/03q8dnn23","country_code":"HK","type":"education","lineage":["https://openalex.org/I168719708"]}],"countries":["HK","US"],"is_corresponding":true,"raw_author_name":"Alain Bensoussan","raw_affiliation_strings":["Naveen Jindal School of Management, University of Texas at Dallas, Richardson, TX 75080-3021 USA","School of Data Science, City University of Hong Kong, Kowloon, Hong Kong SAR"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Naveen Jindal School of Management, University of Texas at Dallas, Richardson, TX 75080-3021 USA","institution_ids":["https://openalex.org/I162577319"]},{"raw_affiliation_string":"School of Data Science, City University of Hong Kong, Kowloon, Hong Kong SAR","institution_ids":["https://openalex.org/I168719708"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5019680632","display_name":"Jiayue Han","orcid":"https://orcid.org/0000-0002-6630-263X"},"institutions":[{"id":"https://openalex.org/I168719708","display_name":"City University of Hong Kong","ror":"https://ror.org/03q8dnn23","country_code":"HK","type":"education","lineage":["https://openalex.org/I168719708"]}],"countries":["HK"],"is_corresponding":false,"raw_author_name":"Jiayue Han","raw_affiliation_strings":["School of Data Science, City University of Hong Kong, Kowloon, Hong Kong SAR"],"raw_orcid":"https://orcid.org/0000-0002-6630-263X","affiliations":[{"raw_affiliation_string":"School of Data Science, City University of Hong Kong, Kowloon, Hong Kong SAR","institution_ids":["https://openalex.org/I168719708"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5075984844","display_name":"Sheung Chi Phillip Yam","orcid":"https://orcid.org/0000-0002-4380-0919"},"institutions":[{"id":"https://openalex.org/I177725633","display_name":"Chinese University of Hong Kong","ror":"https://ror.org/00t33hh48","country_code":"HK","type":"education","lineage":["https://openalex.org/I177725633"]}],"countries":["HK"],"is_corresponding":false,"raw_author_name":"Sheung Chi Phillip Yam","raw_affiliation_strings":["Department of Statistics, Chinese University of Hong Kong, Shatin, NT, Hong Kong SAR"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Statistics, Chinese University of Hong Kong, Shatin, NT, Hong Kong SAR","institution_ids":["https://openalex.org/I177725633"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5030761097","display_name":"Xiang Zhou","orcid":"https://orcid.org/0000-0002-3835-3894"},"institutions":[{"id":"https://openalex.org/I168719708","display_name":"City University of Hong Kong","ror":"https://ror.org/03q8dnn23","country_code":"HK","type":"education","lineage":["https://openalex.org/I168719708"]}],"countries":["HK"],"is_corresponding":false,"raw_author_name":"Xiang Zhou","raw_affiliation_strings":["Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong SAR","School of Data Science, City University of Hong Kong, Kowloon, Hong Kong SAR"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong SAR","institution_ids":["https://openalex.org/I168719708"]},{"raw_affiliation_string":"School of Data Science, City University of Hong Kong, Kowloon, Hong Kong SAR","institution_ids":["https://openalex.org/I168719708"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5027426452"],"corresponding_institution_ids":["https://openalex.org/I162577319","https://openalex.org/I168719708"],"apc_list":null,"apc_paid":null,"fwci":0.6435,"has_fulltext":false,"cited_by_count":4,"citation_normalized_percentile":{"value":0.62982131,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":96},"biblio":{"volume":"61","issue":"2","first_page":"973","last_page":"994"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11206","display_name":"Model Reduction and Neural Networks","score":0.9962000250816345,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11206","display_name":"Model Reduction and Neural Networks","score":0.9962000250816345,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10791","display_name":"Advanced Control Systems Optimization","score":0.9891999959945679,"subfield":{"id":"https://openalex.org/subfields/2207","display_name":"Control and Systems Engineering"},"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/T10339","display_name":"Advanced Numerical Methods in Computational Mathematics","score":0.9708999991416931,"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":0.7163951396942139},{"id":"https://openalex.org/keywords/bellman-equation","display_name":"Bellman equation","score":0.553591787815094},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.517964243888855},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.48809492588043213},{"id":"https://openalex.org/keywords/optimal-control","display_name":"Optimal control","score":0.48720210790634155},{"id":"https://openalex.org/keywords/robustness","display_name":"Robustness (evolution)","score":0.4818342924118042},{"id":"https://openalex.org/keywords/gradient-method","display_name":"Gradient method","score":0.44962817430496216},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.43866121768951416},{"id":"https://openalex.org/keywords/initial-value-problem","display_name":"Initial value problem","score":0.4237591028213501},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.4135803282260895},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.17831823229789734}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7163951396942139},{"id":"https://openalex.org/C14646407","wikidata":"https://www.wikidata.org/wiki/Q1430750","display_name":"Bellman equation","level":2,"score":0.553591787815094},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.517964243888855},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.48809492588043213},{"id":"https://openalex.org/C91575142","wikidata":"https://www.wikidata.org/wiki/Q1971426","display_name":"Optimal control","level":2,"score":0.48720210790634155},{"id":"https://openalex.org/C63479239","wikidata":"https://www.wikidata.org/wiki/Q7353546","display_name":"Robustness (evolution)","level":3,"score":0.4818342924118042},{"id":"https://openalex.org/C115680565","wikidata":"https://www.wikidata.org/wiki/Q5977448","display_name":"Gradient method","level":2,"score":0.44962817430496216},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.43866121768951416},{"id":"https://openalex.org/C26955809","wikidata":"https://www.wikidata.org/wiki/Q10167591","display_name":"Initial value problem","level":2,"score":0.4237591028213501},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.4135803282260895},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.17831823229789734},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"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/C104317684","wikidata":"https://www.wikidata.org/wiki/Q7187","display_name":"Gene","level":2,"score":0.0},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0},{"id":"https://openalex.org/C55493867","wikidata":"https://www.wikidata.org/wiki/Q7094","display_name":"Biochemistry","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1137/21m1442838","is_oa":false,"landing_page_url":"https://doi.org/10.1137/21m1442838","pdf_url":null,"source":{"id":"https://openalex.org/S203348814","display_name":"SIAM Journal on Numerical Analysis","issn_l":"0036-1429","issn":["0036-1429","1095-7170"],"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 Numerical Analysis","raw_type":"journal-article"},{"is_oa":true,"landing_page_url":"https://scholars.cityu.edu.hk/en/publications/value-gradient-based-formulation-of-optimal-control-problem-and-m","pdf_url":null,"source":{"id":"https://openalex.org/S203348814","display_name":"SIAM Journal on Numerical Analysis","issn_l":"0036-1429","issn":["0036-1429","1095-7170"],"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 Numerical Analysis","raw_type":"journal-article"},{"id":"pmh:oai:pure.atira.dk:publications/ef143737-e2e8-4266-aa29-993152843f7a","is_oa":true,"landing_page_url":"https://hdl.handle.net/2031/ef143737-e2e8-4266-aa29-993152843f7a","pdf_url":null,"source":{"id":"https://openalex.org/S7407055387","display_name":"CityU Scholars","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"BENSOUSSAN, A, HAN, J, YAM, S C P & ZHOU, X 2023, 'Value-gradient based formulation of optimal control problem and machine learning algorithm', SIAM Journal on Numerical Analysis, vol. 61, no. 2, pp. 973-994. https://doi.org/10.1137/21M1442838","raw_type":"article"}],"best_oa_location":{"is_oa":true,"landing_page_url":"https://scholars.cityu.edu.hk/en/publications/value-gradient-based-formulation-of-optimal-control-problem-and-m","pdf_url":null,"source":{"id":"https://openalex.org/S203348814","display_name":"SIAM Journal on Numerical Analysis","issn_l":"0036-1429","issn":["0036-1429","1095-7170"],"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 Numerical Analysis","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G2460214095","display_name":null,"funder_award_id":"DMS-1905449","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G4067613758","display_name":null,"funder_award_id":"14301321","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G7983688679","display_name":null,"funder_award_id":"NSF-DMS 2204795","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"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":37,"referenced_works":["https://openalex.org/W132270071","https://openalex.org/W1492186721","https://openalex.org/W1494982173","https://openalex.org/W1991113069","https://openalex.org/W1991927580","https://openalex.org/W2003009438","https://openalex.org/W2028145673","https://openalex.org/W2042680115","https://openalex.org/W2047090868","https://openalex.org/W2068328804","https://openalex.org/W2076337359","https://openalex.org/W2091565802","https://openalex.org/W2098432798","https://openalex.org/W2149713433","https://openalex.org/W2588542536","https://openalex.org/W2605995510","https://openalex.org/W2617807896","https://openalex.org/W2803629276","https://openalex.org/W2804631621","https://openalex.org/W2822752092","https://openalex.org/W2903857590","https://openalex.org/W2959078751","https://openalex.org/W2963395620","https://openalex.org/W2963800981","https://openalex.org/W2964104089","https://openalex.org/W2964179106","https://openalex.org/W2971638107","https://openalex.org/W3007470329","https://openalex.org/W3016378723","https://openalex.org/W3043050416","https://openalex.org/W3082030075","https://openalex.org/W3086293388","https://openalex.org/W3112673444","https://openalex.org/W4205410385","https://openalex.org/W4210840654","https://openalex.org/W4296154082","https://openalex.org/W4302609762"],"related_works":["https://openalex.org/W1859185863","https://openalex.org/W2964126589","https://openalex.org/W4285537323","https://openalex.org/W1932159282","https://openalex.org/W4307691181","https://openalex.org/W3016225735","https://openalex.org/W2017412075","https://openalex.org/W1518875602","https://openalex.org/W2116748936","https://openalex.org/W2767258356"],"abstract_inverted_index":{"Optimal":[0],"control":[1,69],"problem":[2],"is":[3,177],"typically":[4],"solved":[5],"by":[6,84],"first":[7],"finding":[8],"the":[9,13,20,23,27,36,45,48,61,86,91,96,99,103,127,159,170,181,187,204,209,214],"value":[10,37,49,100,160,188],"function":[11,50,161,189],"through":[12],"Hamilton-Jacobi":[14],"equation":[15],"(HJE)":[16],"and":[17,156,162,190,211,231],"then":[18],"taking":[19],"minimizer":[21],"of":[22,33,47,56,63,80,153,213],"Hamiltonian":[24],"to":[25,179],"obtain":[26],"control.":[28],"In":[29],"this":[30,78,109,197],"work,":[31],"instead":[32],"focusing":[34],"on":[35,166],"function,":[38],"we":[39,106,130,141],"propose":[40],"a":[41,53,64,123,132],"new":[42,198],"formulation":[43],"for":[44,77,98,117,185,229],"gradient":[46,164],"(value-gradient)":[51],"as":[52,95,169],"decoupled":[54],"system":[55,79],"partial":[57],"differential":[58],"equations":[59,81],"in":[60,82,114,122],"context":[62],"continuous-time":[65],"deterministic":[66],"discounted":[67],"optimal":[68],"problem.":[70],"We":[71],"develop":[72],"an":[73,151],"efficient":[74],"iterative":[75,110],"scheme":[76,111],"parallel":[83],"utilizing":[85],"fact":[87],"that":[88,108,196],"they":[89],"share":[90],"same":[92],"characteristic":[93,133,144],"curves":[94,145],"HJE":[97],"function.":[101,125],"For":[102,126],"theoretical":[104],"part,":[105],"prove":[107],"converges":[112],"linearly":[113],"<i>L</i><sup>2</sup><sub>\u03b1</sub>":[115],"sense":[116],"some":[118],"suitable":[119],"exponent":[120],"\u03b1":[121],"weight":[124],"numerical":[128,215],"method,":[129],"combine":[131],"line":[134],"method":[135,199],"with":[136,218],"machine":[137,175],"learning":[138,176],"techniques.":[139],"Specifically,":[140],"generate":[142],"multiple":[143],"at":[146],"each":[147,167],"policy":[148],"iteration":[149],"from":[150],"ensemble":[152],"initial":[154],"states":[155],"compute":[157],"both":[158,186],"its":[163,191],"simultaneously":[165],"curve":[168],"labeled":[171],"data.":[172],"Then":[173],"supervised":[174],"applied":[178],"minimize":[180],"weighted":[182],"squared":[183],"loss":[184],"gradients.":[192],"Experimental":[193],"results":[194],"demonstrate":[195],"not":[200],"only":[201],"significantly":[202],"increases":[203],"accuracy":[205],"but":[206],"also":[207],"improves":[208],"efficiency":[210],"robustness":[212],"estimates,":[216],"particularly":[217],"less":[219],"characteristics":[220],"data":[221],"or":[222],"fewer":[223],"training":[224],"steps.":[225],"\u00a9":[226],"2023":[227],"Society":[228],"Industrial":[230],"Applied":[232],"Mathematics":[233]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":2}],"updated_date":"2026-04-29T09:16:38.111599","created_date":"2025-10-10T00:00:00"}