{"id":"https://openalex.org/W4323566651","doi":"https://doi.org/10.3390/a16030148","title":"Properties of the Quadratic Transformation of Dual Variables","display_name":"Properties of the Quadratic Transformation of Dual Variables","publication_year":2023,"publication_date":"2023-03-07","ids":{"openalex":"https://openalex.org/W4323566651","doi":"https://doi.org/10.3390/a16030148"},"language":"en","primary_location":{"id":"doi:10.3390/a16030148","is_oa":true,"landing_page_url":"https://doi.org/10.3390/a16030148","pdf_url":"https://www.mdpi.com/1999-4893/16/3/148/pdf?version=1678191249","source":{"id":"https://openalex.org/S190629608","display_name":"Algorithms","issn_l":"1999-4893","issn":["1999-4893"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Algorithms","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/1999-4893/16/3/148/pdf?version=1678191249","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5088654072","display_name":"V. N. Krutikov","orcid":"https://orcid.org/0000-0001-8704-1670"},"institutions":[{"id":"https://openalex.org/I4853710","display_name":"Siberian Federal University","ror":"https://ror.org/05fw97k56","country_code":"RU","type":"education","lineage":["https://openalex.org/I4853710"]},{"id":"https://openalex.org/I52135307","display_name":"Kemerovo State University","ror":"https://ror.org/036yvre49","country_code":"RU","type":"education","lineage":["https://openalex.org/I52135307"]}],"countries":["RU"],"is_corresponding":false,"raw_author_name":"Vladimir Krutikov","raw_affiliation_strings":["Department of Applied Mathematics, Kemerovo State University, 6 Krasnaya Street, Kemerovo 650043, Russia","Laboratory \u201cHybrid Methods of Modeling and Optimization in Complex Systems\u201d, Siberian Federal University, 79 Prosp. Svobodny, Krasnoyarsk 660041, Russia","Laboratory \"Hybrid Methods of Modeling and Optimization in Complex Systems\", Siberian Federal University, 79 Prosp. Svobodny, Krasnoyarsk 660041, Russia"],"raw_orcid":"https://orcid.org/0000-0001-8704-1670","affiliations":[{"raw_affiliation_string":"Department of Applied Mathematics, Kemerovo State University, 6 Krasnaya Street, Kemerovo 650043, Russia","institution_ids":["https://openalex.org/I52135307"]},{"raw_affiliation_string":"Laboratory \u201cHybrid Methods of Modeling and Optimization in Complex Systems\u201d, Siberian Federal University, 79 Prosp. Svobodny, Krasnoyarsk 660041, Russia","institution_ids":["https://openalex.org/I4853710"]},{"raw_affiliation_string":"Laboratory \"Hybrid Methods of Modeling and Optimization in Complex Systems\", Siberian Federal University, 79 Prosp. Svobodny, Krasnoyarsk 660041, Russia","institution_ids":["https://openalex.org/I4853710"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5051694825","display_name":"Elena Tovbis","orcid":"https://orcid.org/0000-0002-8500-2050"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Elena Tovbis","raw_affiliation_strings":["Institite of Informatics and Telecommunications, Reshetnev Siberian State University of Science and Technology, 31 Krasnoyarskii Rabochii Prospekt, Krasnoyarsk 660037, Russia"],"raw_orcid":"https://orcid.org/0000-0002-8500-2050","affiliations":[{"raw_affiliation_string":"Institite of Informatics and Telecommunications, Reshetnev Siberian State University of Science and Technology, 31 Krasnoyarskii Rabochii Prospekt, Krasnoyarsk 660037, Russia","institution_ids":[]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5103871043","display_name":"Anatoly Bykov","orcid":null},"institutions":[{"id":"https://openalex.org/I4210137651","display_name":"Federal Research Center for Information and Computational Technologies","ror":"https://ror.org/02x3mf211","country_code":"RU","type":"facility","lineage":["https://openalex.org/I4210137651"]}],"countries":["RU"],"is_corresponding":false,"raw_author_name":"Anatoly Bykov","raw_affiliation_strings":["Institute of Computational Technologies SB RAS, 21 Rukavishnikov Street, Kemerovo 650025, Russia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute of Computational Technologies SB RAS, 21 Rukavishnikov Street, Kemerovo 650025, Russia","institution_ids":["https://openalex.org/I4210137651"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5038736735","display_name":"Predrag S. Stanimirovi\u0107","orcid":"https://orcid.org/0000-0003-0655-3741"},"institutions":[{"id":"https://openalex.org/I152518017","display_name":"University of Nis","ror":"https://ror.org/00965bg92","country_code":"RS","type":"education","lineage":["https://openalex.org/I152518017"]},{"id":"https://openalex.org/I4853710","display_name":"Siberian Federal University","ror":"https://ror.org/05fw97k56","country_code":"RU","type":"education","lineage":["https://openalex.org/I4853710"]}],"countries":["RS","RU"],"is_corresponding":false,"raw_author_name":"Predrag Stanimirovic","raw_affiliation_strings":["Faculty of Sciences and Mathematics, University of Nis, 18000 Nis, Serbia","Laboratory \u201cHybrid Methods of Modeling and Optimization in Complex Systems\u201d, Siberian Federal University, 79 Prosp. Svobodny, Krasnoyarsk 660041, Russia","Laboratory \"Hybrid Methods of Modeling and Optimization in Complex Systems\", Siberian Federal University, 79 Prosp. Svobodny, Krasnoyarsk 660041, Russia"],"raw_orcid":"https://orcid.org/0000-0003-0655-3741","affiliations":[{"raw_affiliation_string":"Faculty of Sciences and Mathematics, University of Nis, 18000 Nis, Serbia","institution_ids":["https://openalex.org/I152518017"]},{"raw_affiliation_string":"Laboratory \u201cHybrid Methods of Modeling and Optimization in Complex Systems\u201d, Siberian Federal University, 79 Prosp. Svobodny, Krasnoyarsk 660041, Russia","institution_ids":["https://openalex.org/I4853710"]},{"raw_affiliation_string":"Laboratory \"Hybrid Methods of Modeling and Optimization in Complex Systems\", Siberian Federal University, 79 Prosp. Svobodny, Krasnoyarsk 660041, Russia","institution_ids":["https://openalex.org/I4853710"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5066806014","display_name":"E S Chernova","orcid":"https://orcid.org/0000-0003-4613-2773"},"institutions":[{"id":"https://openalex.org/I52135307","display_name":"Kemerovo State University","ror":"https://ror.org/036yvre49","country_code":"RU","type":"education","lineage":["https://openalex.org/I52135307"]}],"countries":["RU"],"is_corresponding":false,"raw_author_name":"Ekaterina Chernova","raw_affiliation_strings":["Department of Applied Mathematics, Kemerovo State University, 6 Krasnaya Street, Kemerovo 650043, Russia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Applied Mathematics, Kemerovo State University, 6 Krasnaya Street, Kemerovo 650043, Russia","institution_ids":["https://openalex.org/I52135307"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5081398489","display_name":"Lev Kazakovtsev","orcid":"https://orcid.org/0000-0002-0667-4001"},"institutions":[{"id":"https://openalex.org/I4853710","display_name":"Siberian Federal University","ror":"https://ror.org/05fw97k56","country_code":"RU","type":"education","lineage":["https://openalex.org/I4853710"]}],"countries":["RU"],"is_corresponding":true,"raw_author_name":"Lev Kazakovtsev","raw_affiliation_strings":["Institite of Informatics and Telecommunications, Reshetnev Siberian State University of Science and Technology, 31 Krasnoyarskii Rabochii Prospekt, Krasnoyarsk 660037, Russia","Laboratory \u201cHybrid Methods of Modeling and Optimization in Complex Systems\u201d, Siberian Federal University, 79 Prosp. Svobodny, Krasnoyarsk 660041, Russia","Laboratory \"Hybrid Methods of Modeling and Optimization in Complex Systems\", Siberian Federal University, 79 Prosp. Svobodny, Krasnoyarsk 660041, Russia"],"raw_orcid":"https://orcid.org/0000-0002-0667-4001","affiliations":[{"raw_affiliation_string":"Institite of Informatics and Telecommunications, Reshetnev Siberian State University of Science and Technology, 31 Krasnoyarskii Rabochii Prospekt, Krasnoyarsk 660037, Russia","institution_ids":[]},{"raw_affiliation_string":"Laboratory \u201cHybrid Methods of Modeling and Optimization in Complex Systems\u201d, Siberian Federal University, 79 Prosp. Svobodny, Krasnoyarsk 660041, Russia","institution_ids":["https://openalex.org/I4853710"]},{"raw_affiliation_string":"Laboratory \"Hybrid Methods of Modeling and Optimization in Complex Systems\", Siberian Federal University, 79 Prosp. Svobodny, Krasnoyarsk 660041, Russia","institution_ids":["https://openalex.org/I4853710"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":6,"corresponding_author_ids":["https://openalex.org/A5081398489"],"corresponding_institution_ids":["https://openalex.org/I4853710"],"apc_list":{"value":1400,"currency":"CHF","value_usd":1515},"apc_paid":{"value":1400,"currency":"CHF","value_usd":1515},"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.04179179,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"16","issue":"3","first_page":"148","last_page":"148"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.987500011920929,"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":0.987500011920929,"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":0.9736999869346619,"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/T11502","display_name":"Facility Location and Emergency Management","score":0.9603999853134155,"subfield":{"id":"https://openalex.org/subfields/1407","display_name":"Organizational Behavior and Human Resource Management"},"field":{"id":"https://openalex.org/fields/14","display_name":"Business, Management and Accounting"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6906532049179077},{"id":"https://openalex.org/keywords/subgradient-method","display_name":"Subgradient method","score":0.6434540152549744},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.542923629283905},{"id":"https://openalex.org/keywords/transformation","display_name":"Transformation (genetics)","score":0.4905490279197693},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4885424077510834},{"id":"https://openalex.org/keywords/quadratic-programming","display_name":"Quadratic programming","score":0.48750540614128113},{"id":"https://openalex.org/keywords/quadratic-function","display_name":"Quadratic function","score":0.4659968614578247},{"id":"https://openalex.org/keywords/relaxation","display_name":"Relaxation (psychology)","score":0.4570082128047943},{"id":"https://openalex.org/keywords/quadratic-equation","display_name":"Quadratic equation","score":0.4549729824066162},{"id":"https://openalex.org/keywords/gradient-method","display_name":"Gradient method","score":0.43013542890548706},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.41061389446258545},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.33731672167778015}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6906532049179077},{"id":"https://openalex.org/C158968445","wikidata":"https://www.wikidata.org/wiki/Q7631150","display_name":"Subgradient method","level":2,"score":0.6434540152549744},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.542923629283905},{"id":"https://openalex.org/C204241405","wikidata":"https://www.wikidata.org/wiki/Q461499","display_name":"Transformation (genetics)","level":3,"score":0.4905490279197693},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4885424077510834},{"id":"https://openalex.org/C81845259","wikidata":"https://www.wikidata.org/wiki/Q290117","display_name":"Quadratic programming","level":2,"score":0.48750540614128113},{"id":"https://openalex.org/C166437778","wikidata":"https://www.wikidata.org/wiki/Q50695","display_name":"Quadratic function","level":3,"score":0.4659968614578247},{"id":"https://openalex.org/C2776029896","wikidata":"https://www.wikidata.org/wiki/Q3935810","display_name":"Relaxation (psychology)","level":2,"score":0.4570082128047943},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.4549729824066162},{"id":"https://openalex.org/C115680565","wikidata":"https://www.wikidata.org/wiki/Q5977448","display_name":"Gradient method","level":2,"score":0.43013542890548706},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.41061389446258545},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.33731672167778015},{"id":"https://openalex.org/C55493867","wikidata":"https://www.wikidata.org/wiki/Q7094","display_name":"Biochemistry","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},{"id":"https://openalex.org/C15744967","wikidata":"https://www.wikidata.org/wiki/Q9418","display_name":"Psychology","level":0,"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/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","level":1,"score":0.0},{"id":"https://openalex.org/C77805123","wikidata":"https://www.wikidata.org/wiki/Q161272","display_name":"Social psychology","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/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/a16030148","is_oa":true,"landing_page_url":"https://doi.org/10.3390/a16030148","pdf_url":"https://www.mdpi.com/1999-4893/16/3/148/pdf?version=1678191249","source":{"id":"https://openalex.org/S190629608","display_name":"Algorithms","issn_l":"1999-4893","issn":["1999-4893"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Algorithms","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:e6dfa253d9e44ba2b6db74090874c7c4","is_oa":true,"landing_page_url":"https://doaj.org/article/e6dfa253d9e44ba2b6db74090874c7c4","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","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":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Algorithms, Vol 16, Iss 3, p 148 (2023)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/1999-4893/16/3/148/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/a16030148","pdf_url":null,"source":{"id":"https://openalex.org/S4306400947","display_name":"MDPI (MDPI AG)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210097602","host_organization_name":"Multidisciplinary Digital Publishing Institute (Switzerland)","host_organization_lineage":["https://openalex.org/I4210097602"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Algorithms; Volume 16; Issue 3; Pages: 148","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/a16030148","is_oa":true,"landing_page_url":"https://doi.org/10.3390/a16030148","pdf_url":"https://www.mdpi.com/1999-4893/16/3/148/pdf?version=1678191249","source":{"id":"https://openalex.org/S190629608","display_name":"Algorithms","issn_l":"1999-4893","issn":["1999-4893"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Algorithms","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G5057159857","display_name":null,"funder_award_id":"075-15-2022-1121","funder_id":"https://openalex.org/F4320327494","funder_display_name":"Ministry of Science and Higher Education of the Russian Federation"},{"id":"https://openalex.org/G597338630","display_name":null,"funder_award_id":"075-15-2022","funder_id":"https://openalex.org/F4320327494","funder_display_name":"Ministry of Science and Higher Education of the Russian Federation"},{"id":"https://openalex.org/G6444257319","display_name":null,"funder_award_id":"075-15-","funder_id":"https://openalex.org/F4320327494","funder_display_name":"Ministry of Science and Higher Education of the Russian Federation"}],"funders":[{"id":"https://openalex.org/F4320327494","display_name":"Ministry of Science and Higher Education of the Russian Federation","ror":null}],"has_content":{"grobid_xml":false,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4323566651.pdf"},"referenced_works_count":53,"referenced_works":["https://openalex.org/W11112849","https://openalex.org/W1638543427","https://openalex.org/W1922442141","https://openalex.org/W2005136695","https://openalex.org/W2038210983","https://openalex.org/W2073403745","https://openalex.org/W2078409719","https://openalex.org/W2084229671","https://openalex.org/W2103111465","https://openalex.org/W2133503298","https://openalex.org/W2140180727","https://openalex.org/W2169713291","https://openalex.org/W2510415425","https://openalex.org/W2587454369","https://openalex.org/W2752013009","https://openalex.org/W2913752119","https://openalex.org/W2923772752","https://openalex.org/W2963433607","https://openalex.org/W2963843970","https://openalex.org/W2994571485","https://openalex.org/W3010541578","https://openalex.org/W3017512408","https://openalex.org/W3043674922","https://openalex.org/W3092479710","https://openalex.org/W3100569850","https://openalex.org/W3120050151","https://openalex.org/W3129313077","https://openalex.org/W3170092489","https://openalex.org/W3176377952","https://openalex.org/W3184430965","https://openalex.org/W3194305617","https://openalex.org/W3197885837","https://openalex.org/W3198274828","https://openalex.org/W3199086593","https://openalex.org/W3201914388","https://openalex.org/W3209336782","https://openalex.org/W3211250483","https://openalex.org/W3212136124","https://openalex.org/W4206450434","https://openalex.org/W4211092921","https://openalex.org/W4223509507","https://openalex.org/W4240463582","https://openalex.org/W4252076505","https://openalex.org/W4295883465","https://openalex.org/W4297147019","https://openalex.org/W4303521013","https://openalex.org/W4307550684","https://openalex.org/W4308168231","https://openalex.org/W4309267112","https://openalex.org/W4309778145","https://openalex.org/W4312334033","https://openalex.org/W4320496126","https://openalex.org/W6803830308"],"related_works":["https://openalex.org/W269925151","https://openalex.org/W3036464408","https://openalex.org/W2020990755","https://openalex.org/W3096788047","https://openalex.org/W2377851227","https://openalex.org/W2101513891","https://openalex.org/W4293088966","https://openalex.org/W2366868484","https://openalex.org/W4386731488","https://openalex.org/W2364127131"],"abstract_inverted_index":{"We":[0,31,51,72,160,180],"investigate":[1,52],"a":[2,5,10,152,188,192,215,219,224,232],"solution":[3,233],"of":[4,29,37,39,57,61,67,78,85,88,103,110,114,126,155,157,166,169,175,195,234,262,277,280,287],"convex":[6,12],"programming":[7],"problem":[8,23,236],"with":[9,151,191,252],"strongly":[11],"objective":[13],"function":[14,90,118,128,186,243],"based":[15],"on":[16,26],"the":[17,27,33,53,59,65,76,80,83,86,89,95,101,104,107,111,115,121,124,127,164,167,173,183,207,235,268,273,281],"dual":[18,21,40,62,117,170,185,263],"approach.":[19],"A":[20],"optimization":[22,49,149],"has":[24,187],"constraints":[25],"positivity":[28],"variables.":[30],"study":[32],"methods":[34,136,204,240,245],"and":[35,120,143,241,249,255],"properties":[36,211],"transformations":[38,261],"variables":[41,63,119,171,264],"that":[42,74,138,144,182],"enable":[43],"us":[44,199],"to":[45,132,141,162,200,205,223,230],"obtain":[46,231],"an":[47],"unconstrained":[48],"problem.":[50],"previously":[54],"known":[55],"method":[56,194,248],"transforming":[58],"components":[60,168],"in":[64,75,106,123,172,267],"form":[66,174],"their":[68,176],"modulus":[69,81,225],"(modulus":[70],"method).":[71,179],"show":[73],"case":[77],"using":[79],"method,":[82],"degree":[84,125,154],"degeneracy":[87],"increases":[91],"as":[92],"it":[93,227],"approaches":[94],"optimal":[96],"point.":[97],"Taking":[98],"into":[99],"account":[100],"ambiguity":[102],"gradient":[105,190,203,247],"boundary":[108],"regions":[109],"sign":[112],"change":[113],"new":[116],"increase":[122],"degeneracy,":[129],"we":[130],"need":[131],"use":[133,163,201],"relaxation":[134,238],"subgradient":[135,239],"(RSM)":[137],"are":[139,212],"difficult":[140],"implement":[142],"can":[145],"solve":[146],"non-smooth":[147],"non-convex":[148],"problems":[150],"high":[153],"elongation":[156],"level":[158],"surfaces.":[159],"propose":[161],"transformation":[165,221],"square":[177],"(quadratic":[178],"prove":[181],"transformed":[184],"Lipschitz":[189],"quadratic":[193,220],"transformation.":[196],"This":[197],"enables":[198],"efficient":[202],"find":[206],"extremum.":[208],"The":[209,259],"above":[210],"confirmed":[213],"by":[214,237],"computational":[216,257],"experiment.":[217],"With":[218],"compared":[222],"transformation,":[226],"is":[228],"possible":[229],"smooth":[242],"minimization":[244],"(conjugate":[246],"quasi-Newtonian":[250],"method)":[251],"higher":[253],"accuracy":[254],"lower":[256],"costs.":[258],"noted":[260],"were":[265],"used":[266],"program":[269],"module":[270],"for":[271,284],"calculating":[272],"maximum":[274],"permissible":[275],"emissions":[276],"enterprises":[278],"(MPE)":[279],"software":[282],"package":[283],"environmental":[285],"monitoring":[286],"atmospheric":[288],"air":[289],"(ERA-AIR).":[290]},"counts_by_year":[],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2023-03-09T00:00:00"}