{"id":"https://openalex.org/W2566453305","doi":"https://doi.org/10.1080/10556788.2016.1266354","title":"An efficient modified Polak\u2013Ribi\u00e8re\u2013Polyak conjugate gradient method with global convergence properties","display_name":"An efficient modified Polak\u2013Ribi\u00e8re\u2013Polyak conjugate gradient method with global convergence properties","publication_year":2016,"publication_date":"2016-12-13","ids":{"openalex":"https://openalex.org/W2566453305","doi":"https://doi.org/10.1080/10556788.2016.1266354","mag":"2566453305"},"language":"en","primary_location":{"id":"doi:10.1080/10556788.2016.1266354","is_oa":false,"landing_page_url":"https://doi.org/10.1080/10556788.2016.1266354","pdf_url":null,"source":{"id":"https://openalex.org/S103047102","display_name":"Optimization methods & software","issn_l":"1026-7670","issn":["1026-7670","1029-4937","1055-6788"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320547","host_organization_name":"Taylor & Francis","host_organization_lineage":["https://openalex.org/P4310320547"],"host_organization_lineage_names":["Taylor & Francis"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Optimization Methods and Software","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/A5071272316","display_name":"Ahmad Alhawarat","orcid":"https://orcid.org/0000-0002-8283-0432"},"institutions":[{"id":"https://openalex.org/I92449659","display_name":"Isra University","ror":"https://ror.org/04d4bt482","country_code":"JO","type":"education","lineage":["https://openalex.org/I92449659"]},{"id":"https://openalex.org/I157696592","display_name":"Universiti Malaysia Terengganu","ror":"https://ror.org/02474f074","country_code":"MY","type":"education","lineage":["https://openalex.org/I157696592"]}],"countries":["JO","MY"],"is_corresponding":true,"raw_author_name":"Ahmad Alhawarat","raw_affiliation_strings":["Department of Mathematics, College of Science, Isra University, Amman, Jordan","School of Informatics and Applied Mathematics, Universiti Malaysia Terengganu, 21030 Kuala Terengganu, Terengganu, Malaysia"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, College of Science, Isra University, Amman, Jordan","institution_ids":["https://openalex.org/I92449659"]},{"raw_affiliation_string":"School of Informatics and Applied Mathematics, Universiti Malaysia Terengganu, 21030 Kuala Terengganu, Terengganu, Malaysia","institution_ids":["https://openalex.org/I157696592"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5066635090","display_name":"Zabidin Salleh","orcid":"https://orcid.org/0000-0001-5877-9051"},"institutions":[{"id":"https://openalex.org/I157696592","display_name":"Universiti Malaysia Terengganu","ror":"https://ror.org/02474f074","country_code":"MY","type":"education","lineage":["https://openalex.org/I157696592"]}],"countries":["MY"],"is_corresponding":false,"raw_author_name":"Zabidin Salleh","raw_affiliation_strings":["School of Informatics and Applied Mathematics, Universiti Malaysia Terengganu, 21030 Kuala Terengganu, Terengganu, Malaysia"],"affiliations":[{"raw_affiliation_string":"School of Informatics and Applied Mathematics, Universiti Malaysia Terengganu, 21030 Kuala Terengganu, Terengganu, Malaysia","institution_ids":["https://openalex.org/I157696592"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5014736688","display_name":"Mustafa Mamat\uf020","orcid":"https://orcid.org/0000-0002-4802-3733"},"institutions":[{"id":"https://openalex.org/I51569856","display_name":"Sultan Zainal Abidin University","ror":"https://ror.org/00bnk2e50","country_code":"MY","type":"education","lineage":["https://openalex.org/I51569856"]}],"countries":["MY"],"is_corresponding":false,"raw_author_name":"Mustafa Mamat","raw_affiliation_strings":["Faculty of Informatics and Computing, Universiti Sultan Zainal Abidin, Kuala Terengganu, Malaysia"],"affiliations":[{"raw_affiliation_string":"Faculty of Informatics and Computing, Universiti Sultan Zainal Abidin, Kuala Terengganu, Malaysia","institution_ids":["https://openalex.org/I51569856"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5053756192","display_name":"Mohd Rivaie","orcid":"https://orcid.org/0000-0001-5481-8981"},"institutions":[{"id":"https://openalex.org/I157696592","display_name":"Universiti Malaysia Terengganu","ror":"https://ror.org/02474f074","country_code":"MY","type":"education","lineage":["https://openalex.org/I157696592"]}],"countries":["MY"],"is_corresponding":false,"raw_author_name":"Mohd Rivaie","raw_affiliation_strings":["Department of Computer Science and Mathematics, Universiti Technologi MARA (UITM) Terengganu, Campus Kuala Terengganu, Malaysia"],"affiliations":[{"raw_affiliation_string":"Department of Computer Science and Mathematics, Universiti Technologi MARA (UITM) Terengganu, Campus Kuala Terengganu, Malaysia","institution_ids":["https://openalex.org/I157696592"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5071272316"],"corresponding_institution_ids":["https://openalex.org/I157696592","https://openalex.org/I92449659"],"apc_list":null,"apc_paid":null,"fwci":2.3028,"has_fulltext":false,"cited_by_count":26,"citation_normalized_percentile":{"value":0.88130147,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":100},"biblio":{"volume":"32","issue":"6","first_page":"1299","last_page":"1312"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.9998999834060669,"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":0.9998999834060669,"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":0.991599977016449,"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/T12661","display_name":"Iterative Methods for Nonlinear Equations","score":0.9883999824523926,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/line-search","display_name":"Line search","score":0.8060232996940613},{"id":"https://openalex.org/keywords/conjugate-gradient-method","display_name":"Conjugate gradient method","score":0.7875698804855347},{"id":"https://openalex.org/keywords/lipschitz-continuity","display_name":"Lipschitz continuity","score":0.6539537310600281},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6498538255691528},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.6318525671958923},{"id":"https://openalex.org/keywords/nonlinear-conjugate-gradient-method","display_name":"Nonlinear conjugate gradient method","score":0.5306612253189087},{"id":"https://openalex.org/keywords/gradient-descent","display_name":"Gradient descent","score":0.51987624168396},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.5138778686523438},{"id":"https://openalex.org/keywords/line","display_name":"Line (geometry)","score":0.49959444999694824},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.47359851002693176},{"id":"https://openalex.org/keywords/bounded-function","display_name":"Bounded function","score":0.4668339788913727},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.4665394127368927},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.32539933919906616},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.12051591277122498},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.06670823693275452},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.06269729137420654}],"concepts":[{"id":"https://openalex.org/C85522705","wikidata":"https://www.wikidata.org/wiki/Q3278015","display_name":"Line search","level":3,"score":0.8060232996940613},{"id":"https://openalex.org/C81184566","wikidata":"https://www.wikidata.org/wiki/Q1191895","display_name":"Conjugate gradient method","level":2,"score":0.7875698804855347},{"id":"https://openalex.org/C22324862","wikidata":"https://www.wikidata.org/wiki/Q652707","display_name":"Lipschitz continuity","level":2,"score":0.6539537310600281},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6498538255691528},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.6318525671958923},{"id":"https://openalex.org/C26362088","wikidata":"https://www.wikidata.org/wiki/Q17086453","display_name":"Nonlinear conjugate gradient method","level":4,"score":0.5306612253189087},{"id":"https://openalex.org/C153258448","wikidata":"https://www.wikidata.org/wiki/Q1199743","display_name":"Gradient descent","level":3,"score":0.51987624168396},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.5138778686523438},{"id":"https://openalex.org/C198352243","wikidata":"https://www.wikidata.org/wiki/Q37105","display_name":"Line (geometry)","level":2,"score":0.49959444999694824},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.47359851002693176},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.4668339788913727},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.4665394127368927},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.32539933919906616},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.12051591277122498},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.06670823693275452},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.06269729137420654},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C178635117","wikidata":"https://www.wikidata.org/wiki/Q747499","display_name":"RADIUS","level":2,"score":0.0},{"id":"https://openalex.org/C38652104","wikidata":"https://www.wikidata.org/wiki/Q3510521","display_name":"Computer security","level":1,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C50522688","wikidata":"https://www.wikidata.org/wiki/Q189833","display_name":"Economic growth","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1080/10556788.2016.1266354","is_oa":false,"landing_page_url":"https://doi.org/10.1080/10556788.2016.1266354","pdf_url":null,"source":{"id":"https://openalex.org/S103047102","display_name":"Optimization methods & software","issn_l":"1026-7670","issn":["1026-7670","1029-4937","1055-6788"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320547","host_organization_name":"Taylor & Francis","host_organization_lineage":["https://openalex.org/P4310320547"],"host_organization_lineage_names":["Taylor & Francis"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Optimization Methods and Software","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G8469824831","display_name":null,"funder_award_id":"59256","funder_id":"https://openalex.org/F4320321709","funder_display_name":"Ministry of Higher Education, Malaysia"}],"funders":[{"id":"https://openalex.org/F4320321709","display_name":"Ministry of Higher Education, Malaysia","ror":"https://ror.org/05mcs2t73"},{"id":"https://openalex.org/F4320322727","display_name":"Minist\u00e8re de l'Education Nationale, de l'Enseignement Superieur et de la Recherche","ror":"https://ror.org/03sjk9a61"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":21,"referenced_works":["https://openalex.org/W89654658","https://openalex.org/W150175414","https://openalex.org/W1968553199","https://openalex.org/W1991360534","https://openalex.org/W2008374458","https://openalex.org/W2012231377","https://openalex.org/W2018215034","https://openalex.org/W2019460058","https://openalex.org/W2028736557","https://openalex.org/W2043382734","https://openalex.org/W2051434435","https://openalex.org/W2065678922","https://openalex.org/W2067693696","https://openalex.org/W2098528027","https://openalex.org/W2108560279","https://openalex.org/W2149454052","https://openalex.org/W2158894942","https://openalex.org/W2230977749","https://openalex.org/W2316564661","https://openalex.org/W2363674510","https://openalex.org/W4254522940"],"related_works":["https://openalex.org/W2098528027","https://openalex.org/W3107945686","https://openalex.org/W2356977358","https://openalex.org/W2384571804","https://openalex.org/W2369191856","https://openalex.org/W3080334989","https://openalex.org/W2355738879","https://openalex.org/W2371381925","https://openalex.org/W1973021988","https://openalex.org/W2350706414"],"abstract_inverted_index":{"The":[0,50,108],"conjugate":[1],"gradient":[2],"(CG)":[3],"method":[4],"is":[5,37,71,80],"one":[6],"of":[7,25,75,103,114],"the":[8,26,58,65,83,90,112,115,121],"most":[9],"popular":[10],"methods":[11],"for":[12,39,82],"solving":[13],"large-scale":[14],"unconstrained":[15,105],"optimization":[16,106],"problems.":[17,107],"In":[18],"this":[19],"paper,":[20],"a":[21,47,76,101],"new":[22,51],"modified":[23],"version":[24],"CG":[27,96,117,123],"formula":[28],"that":[29,41],"was":[30],"introduced":[31],"by":[32],"Polak,":[33],"Ribi\u00e8re,":[34],"and":[35,45,93],"Polyak":[36],"proposed":[38,91,116],"problems":[40],"are":[42,98],"bounded":[43],"below":[44],"have":[46],"Lipschitz-continuous":[48],"gradient.":[49],"parameter":[52,92,118],"provides":[53],"global":[54],"convergence":[55],"properties":[56],"when":[57],"strong":[59],"Wolfe-Powell":[60,67],"(SWP)":[61],"line":[62,69,85],"search":[63,70],"or":[64],"weak":[66],"(WWP)":[68],"employed.":[72],"A":[73],"proof":[74],"sufficient":[77],"descent":[78],"condition":[79],"provided":[81],"SWP":[84],"search.":[86],"Numerical":[87],"comparisons":[88],"between":[89],"other":[94,122],"recent":[95],"modifications":[97],"made":[99],"on":[100],"set":[102],"standard":[104],"numerical":[109],"results":[110],"demonstrate":[111],"efficiency":[113],"compared":[119],"with":[120],"parameters.":[124]},"counts_by_year":[{"year":2026,"cited_by_count":3},{"year":2025,"cited_by_count":6},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":8},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":2},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":2}],"updated_date":"2026-04-10T15:06:20.359241","created_date":"2025-10-10T00:00:00"}
