{"id":"https://openalex.org/W3153264386","doi":"https://doi.org/10.3390/e23040465","title":"On the Locally Polynomial Complexity of the Projection-Gradient Method for Solving Piecewise Quadratic Optimisation Problems","display_name":"On the Locally Polynomial Complexity of the Projection-Gradient Method for Solving Piecewise Quadratic Optimisation Problems","publication_year":2021,"publication_date":"2021-04-15","ids":{"openalex":"https://openalex.org/W3153264386","doi":"https://doi.org/10.3390/e23040465","mag":"3153264386","pmid":"https://pubmed.ncbi.nlm.nih.gov/33920784"},"language":"en","primary_location":{"id":"doi:10.3390/e23040465","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e23040465","pdf_url":"https://www.mdpi.com/1099-4300/23/4/465/pdf?version=1618469864","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"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":"Entropy","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj","pubmed"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/1099-4300/23/4/465/pdf?version=1618469864","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5045768571","display_name":"Agnieszka Prusi\u0144ska","orcid":"https://orcid.org/0000-0002-6091-6884"},"institutions":[{"id":"https://openalex.org/I231086526","display_name":"University of Siedlce","ror":"https://ror.org/01wkb9987","country_code":"PL","type":"education","lineage":["https://openalex.org/I231086526"]}],"countries":["PL"],"is_corresponding":true,"raw_author_name":"Agnieszka Prusi\u0144ska","raw_affiliation_strings":["Faculty of Exact and Natural Sciences, Siedlce University, 08-110 Siedlce, Poland"],"raw_orcid":"https://orcid.org/0000-0002-6091-6884","affiliations":[{"raw_affiliation_string":"Faculty of Exact and Natural Sciences, Siedlce University, 08-110 Siedlce, Poland","institution_ids":["https://openalex.org/I231086526"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5033162198","display_name":"Krzysztof Szkatu\u0142a","orcid":"https://orcid.org/0000-0002-0265-6033"},"institutions":[{"id":"https://openalex.org/I231086526","display_name":"University of Siedlce","ror":"https://ror.org/01wkb9987","country_code":"PL","type":"education","lineage":["https://openalex.org/I231086526"]},{"id":"https://openalex.org/I66083562","display_name":"Systems Research Institute","ror":"https://ror.org/0111cp837","country_code":"PL","type":"facility","lineage":["https://openalex.org/I66083562","https://openalex.org/I99542240"]},{"id":"https://openalex.org/I99542240","display_name":"Polish Academy of Sciences","ror":"https://ror.org/01dr6c206","country_code":"PL","type":"government","lineage":["https://openalex.org/I99542240"]}],"countries":["PL"],"is_corresponding":false,"raw_author_name":"Krzysztof Szkatu\u0142a","raw_affiliation_strings":["Faculty of Exact and Natural Sciences, Siedlce University, 08-110 Siedlce, Poland","Systems Research Institute, Polish Academy of Sciences, 01-447 Warsaw, Poland"],"raw_orcid":"https://orcid.org/0000-0002-0265-6033","affiliations":[{"raw_affiliation_string":"Faculty of Exact and Natural Sciences, Siedlce University, 08-110 Siedlce, Poland","institution_ids":["https://openalex.org/I231086526"]},{"raw_affiliation_string":"Systems Research Institute, Polish Academy of Sciences, 01-447 Warsaw, Poland","institution_ids":["https://openalex.org/I66083562","https://openalex.org/I99542240"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5091663568","display_name":"\u0410. \u0410. \u0422retyakov","orcid":"https://orcid.org/0000-0003-3474-8458"},"institutions":[{"id":"https://openalex.org/I231086526","display_name":"University of Siedlce","ror":"https://ror.org/01wkb9987","country_code":"PL","type":"education","lineage":["https://openalex.org/I231086526"]},{"id":"https://openalex.org/I2801535751","display_name":"Dorodnitsyn Computing Centre","ror":"https://ror.org/01vgda714","country_code":"RU","type":"facility","lineage":["https://openalex.org/I1313323035","https://openalex.org/I2801535751","https://openalex.org/I4210124601"]},{"id":"https://openalex.org/I66083562","display_name":"Systems Research Institute","ror":"https://ror.org/0111cp837","country_code":"PL","type":"facility","lineage":["https://openalex.org/I66083562","https://openalex.org/I99542240"]},{"id":"https://openalex.org/I99542240","display_name":"Polish Academy of Sciences","ror":"https://ror.org/01dr6c206","country_code":"PL","type":"government","lineage":["https://openalex.org/I99542240"]}],"countries":["PL","RU"],"is_corresponding":false,"raw_author_name":"Alexey Tret\u2019yakov","raw_affiliation_strings":["Dorodnicyn Computing Centre of FRC CSC, Russian Academy of Sciences, 119333 Moscow, Russia","Faculty of Exact and Natural Sciences, Siedlce University, 08-110 Siedlce, Poland","Systems Research Institute, Polish Academy of Sciences, 01-447 Warsaw, Poland"],"raw_orcid":"https://orcid.org/0000-0003-3474-8458","affiliations":[{"raw_affiliation_string":"Dorodnicyn Computing Centre of FRC CSC, Russian Academy of Sciences, 119333 Moscow, Russia","institution_ids":["https://openalex.org/I2801535751"]},{"raw_affiliation_string":"Faculty of Exact and Natural Sciences, Siedlce University, 08-110 Siedlce, Poland","institution_ids":["https://openalex.org/I231086526"]},{"raw_affiliation_string":"Systems Research Institute, Polish Academy of Sciences, 01-447 Warsaw, Poland","institution_ids":["https://openalex.org/I66083562","https://openalex.org/I99542240"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5045768571"],"corresponding_institution_ids":["https://openalex.org/I231086526"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.07567444,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"23","issue":"4","first_page":"465","last_page":"465"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.9968000054359436,"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":0.9968000054359436,"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/T12661","display_name":"Iterative Methods for Nonlinear Equations","score":0.9922000169754028,"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/T10792","display_name":"Matrix Theory and Algorithms","score":0.9564999938011169,"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/piecewise","display_name":"Piecewise","score":0.7563683986663818},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.715438723564148},{"id":"https://openalex.org/keywords/quadratic-equation","display_name":"Quadratic equation","score":0.6079866290092468},{"id":"https://openalex.org/keywords/projection","display_name":"Projection (relational algebra)","score":0.570978581905365},{"id":"https://openalex.org/keywords/dimension","display_name":"Dimension (graph theory)","score":0.5502583384513855},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.5497697591781616},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.5380496382713318},{"id":"https://openalex.org/keywords/quadratic-function","display_name":"Quadratic function","score":0.5032934546470642},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4970150291919708},{"id":"https://openalex.org/keywords/quadratic-programming","display_name":"Quadratic programming","score":0.4721134901046753},{"id":"https://openalex.org/keywords/computational-complexity-theory","display_name":"Computational complexity theory","score":0.4705798029899597},{"id":"https://openalex.org/keywords/projection-method","display_name":"Projection method","score":0.438959002494812},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.2776777744293213},{"id":"https://openalex.org/keywords/dykstras-projection-algorithm","display_name":"Dykstra's projection algorithm","score":0.17879396677017212},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.156025230884552},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.0948079526424408}],"concepts":[{"id":"https://openalex.org/C164660894","wikidata":"https://www.wikidata.org/wiki/Q2037833","display_name":"Piecewise","level":2,"score":0.7563683986663818},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.715438723564148},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.6079866290092468},{"id":"https://openalex.org/C57493831","wikidata":"https://www.wikidata.org/wiki/Q3134666","display_name":"Projection (relational algebra)","level":2,"score":0.570978581905365},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.5502583384513855},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.5497697591781616},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.5380496382713318},{"id":"https://openalex.org/C166437778","wikidata":"https://www.wikidata.org/wiki/Q50695","display_name":"Quadratic function","level":3,"score":0.5032934546470642},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4970150291919708},{"id":"https://openalex.org/C81845259","wikidata":"https://www.wikidata.org/wiki/Q290117","display_name":"Quadratic programming","level":2,"score":0.4721134901046753},{"id":"https://openalex.org/C179799912","wikidata":"https://www.wikidata.org/wiki/Q205084","display_name":"Computational complexity theory","level":2,"score":0.4705798029899597},{"id":"https://openalex.org/C65557600","wikidata":"https://www.wikidata.org/wiki/Q7249451","display_name":"Projection method","level":3,"score":0.438959002494812},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.2776777744293213},{"id":"https://openalex.org/C202426404","wikidata":"https://www.wikidata.org/wiki/Q5318686","display_name":"Dykstra's projection algorithm","level":2,"score":0.17879396677017212},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.156025230884552},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.0948079526424408},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0}],"mesh":[],"locations_count":6,"locations":[{"id":"doi:10.3390/e23040465","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e23040465","pdf_url":"https://www.mdpi.com/1099-4300/23/4/465/pdf?version=1618469864","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"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":"Entropy","raw_type":"journal-article"},{"id":"pmid:33920784","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/33920784","pdf_url":null,"source":{"id":"https://openalex.org/S4306525036","display_name":"PubMed","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy (Basel, Switzerland)","raw_type":null},{"id":"pmh:oai:doaj.org/article:96b2fd4ef5a34e49ba37b34473650fac","is_oa":true,"landing_page_url":"https://doaj.org/article/96b2fd4ef5a34e49ba37b34473650fac","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":"Entropy, Vol 23, Iss 4, p 465 (2021)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/1099-4300/23/4/465/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/e23040465","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":"Entropy; Volume 23; Issue 4; Pages: 465","raw_type":"Text"},{"id":"pmh:oai:pubmedcentral.nih.gov:8071127","is_oa":true,"landing_page_url":"https://www.ncbi.nlm.nih.gov/pmc/articles/8071127","pdf_url":null,"source":{"id":"https://openalex.org/S2764455111","display_name":"PubMed Central","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"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":"Entropy (Basel)","raw_type":"Text"},{"id":"pmh:oai:repozytorium.uph.edu.pl:11331/3614","is_oa":true,"landing_page_url":"http://hdl.handle.net/11331/3614","pdf_url":null,"source":{"id":"https://openalex.org/S4306401161","display_name":"RepoS (Uniwersytetu Przyrodniczo-Humanistycznego w Siedlcach)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I231086526","host_organization_name":"University of Siedlce","host_organization_lineage":["https://openalex.org/I231086526"],"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":"","raw_type":"Article"}],"best_oa_location":{"id":"doi:10.3390/e23040465","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e23040465","pdf_url":"https://www.mdpi.com/1099-4300/23/4/465/pdf?version=1618469864","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"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":"Entropy","raw_type":"journal-article"},"sustainable_development_goals":[{"score":0.4099999964237213,"display_name":"Reduced inequalities","id":"https://metadata.un.org/sdg/10"}],"awards":[{"id":"https://openalex.org/G2698659905","display_name":null,"funder_award_id":"61/20/B","funder_id":"https://openalex.org/F4320322733","funder_display_name":"Ministerstwo Edukacji i Nauki"}],"funders":[{"id":"https://openalex.org/F4320322733","display_name":"Ministerstwo Edukacji i Nauki","ror":"https://ror.org/05dwvd537"}],"has_content":{"pdf":true,"grobid_xml":false},"content_urls":{"pdf":"https://content.openalex.org/works/W3153264386.pdf"},"referenced_works_count":20,"referenced_works":["https://openalex.org/W31849839","https://openalex.org/W199891759","https://openalex.org/W1986943481","https://openalex.org/W2030370244","https://openalex.org/W2030723843","https://openalex.org/W2032676126","https://openalex.org/W2037328914","https://openalex.org/W2085999286","https://openalex.org/W2088241859","https://openalex.org/W2522761509","https://openalex.org/W2526264695","https://openalex.org/W2563943617","https://openalex.org/W2602662813","https://openalex.org/W2754872025","https://openalex.org/W4249667877","https://openalex.org/W4285719527","https://openalex.org/W6659843205","https://openalex.org/W6671941731","https://openalex.org/W6672161608","https://openalex.org/W6727482787"],"related_works":["https://openalex.org/W2028806697","https://openalex.org/W2364127131","https://openalex.org/W2106144756","https://openalex.org/W2388496637","https://openalex.org/W2155888019","https://openalex.org/W2061897416","https://openalex.org/W4206704743","https://openalex.org/W2911254004","https://openalex.org/W2905186505","https://openalex.org/W4254843719"],"abstract_inverted_index":{"This":[0],"paper":[1],"proposes":[2],"a":[3,16,19],"method":[4,14,31,55],"for":[5,59],"solving":[6,60],"optimisation":[7],"problems":[8],"involving":[9],"piecewise":[10],"quadratic":[11],"functions.":[12],"The":[13],"provides":[15],"solution":[17,52],"in":[18],"finite":[20],"number":[21],"of":[22,28,35,50,63],"iterations,":[23],"and":[24],"the":[25,29,36,41,46,51],"computational":[26],"complexity":[27],"proposed":[30],"is":[32],"locally":[33],"polynomial":[34],"problem":[37],"dimension,":[38],"i.e.,":[39],"if":[40],"initial":[42],"point":[43],"belongs":[44],"to":[45],"sufficiently":[47],"small":[48],"neighbourhood":[49],"set.":[53],"Proposed":[54],"could":[56],"be":[57],"applied":[58],"large":[61],"systems":[62],"linear":[64],"inequalities.":[65]},"counts_by_year":[],"updated_date":"2026-05-22T06:13:13.366637","created_date":"2025-10-10T00:00:00"}