NPROS: A Not So Pure Random Orthogonal search algorithm—A suite of random optimization algorithms driven by reinforcement learning
We live in a world where waves of novel nature-inspired metaheuristic algorithms keep hitting the shore repeatedly. This never-ending surge of new metaheuristic algorithms is overwhelming to the extent that their novelty is being criticized. In this paper, instead of focusing on metaheuristics, we f...
Saved in:
| Published in | Optimization letters Vol. 18; no. 9; pp. 2091 - 2111 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Berlin/Heidelberg
Springer Berlin Heidelberg
01.12.2024
|
| Subjects | |
| Online Access | Get full text |
| ISSN | 1862-4472 1862-4480 |
| DOI | 10.1007/s11590-023-02038-0 |
Cover
| Abstract | We live in a world where waves of novel nature-inspired metaheuristic algorithms keep hitting the shore repeatedly. This never-ending surge of new metaheuristic algorithms is overwhelming to the extent that their novelty is being criticized. In this paper, instead of focusing on metaheuristics, we focus on pure random search algorithms for global optimization. Pure Random Orthogonal Search (PROS) is a recently published random optimization algorithm, which is strikingly simple, involves no parameter tuning, but is very effective in solving global optimization problems. In this paper, we propose a modified version of the PROS algorithm, which injects a flavor of exploitation into the otherwise purely explorative PROS algorithm. Further, the concepts of reinforcement learning are utilized to provide the proposed algorithm the ability to ‘learn’ to take the optimal actions, to find the global optima. The source code of NPROS is publicly available at: https://github.com/Shahul-Rahman/NPROS |
|---|---|
| AbstractList | We live in a world where waves of novel nature-inspired metaheuristic algorithms keep hitting the shore repeatedly. This never-ending surge of new metaheuristic algorithms is overwhelming to the extent that their novelty is being criticized. In this paper, instead of focusing on metaheuristics, we focus on pure random search algorithms for global optimization. Pure Random Orthogonal Search (PROS) is a recently published random optimization algorithm, which is strikingly simple, involves no parameter tuning, but is very effective in solving global optimization problems. In this paper, we propose a modified version of the PROS algorithm, which injects a flavor of exploitation into the otherwise purely explorative PROS algorithm. Further, the concepts of reinforcement learning are utilized to provide the proposed algorithm the ability to ‘learn’ to take the optimal actions, to find the global optima. The source code of NPROS is publicly available at: https://github.com/Shahul-Rahman/NPROS |
| Author | Hameed, A. S. Syed Shahul Rajagopalan, Narendran |
| Author_xml | – sequence: 1 givenname: A. S. Syed Shahul orcidid: 0000-0001-8828-2919 surname: Hameed fullname: Hameed, A. S. Syed Shahul email: shahulshan81@gmail.com organization: Computer Science and Engineering, National Institute of Technology Puducherry – sequence: 2 givenname: Narendran orcidid: 0000-0002-1829-9587 surname: Rajagopalan fullname: Rajagopalan, Narendran organization: Computer Science and Engineering, National Institute of Technology Puducherry |
| BookMark | eNp9kE1OwzAQRi1UJNrCBVj5AgHbSeqEXVXxJ1Vt1cI6cp1x6yqxK9tFKisWHIETchLSBoHEgsVoZvG9T6PXQx1jDSB0SckVJYRfe0rTnESExc2QOIvICerSbMCiJMlI5-fm7Az1vN8QMqA0z7vofTKbTxc3eIgnNuCFxbOdAzwXprQ1nrqwtitrRIU9CCfXWFQr63RY159vH0PsdzoAtgq7Nm-3Qdf6VQRtzW_U49LpFzB4uccOtFHWSajBBFw1pUab1Tk6VaLycPG9--j57vZp9BCNp_ePo-E4kiynIZISpJBLkkMKrFRioDKeCq4kI4OUAZdZuVwqnse5SjgIkTIeJ0QQriDOpSRxH2Vtr3TWeweqkDocvw1O6KqgpDjYLFqbRWOzONosDij7g26droXb_w_FLeSbsFmBKzZ25xqd_j_qC1zHjgU |
| CitedBy_id | crossref_primary_10_1007_s13369_024_09098_z crossref_primary_10_1007_s42979_024_02924_z |
| Cites_doi | 10.1007/s13042-019-01053-x 10.1007/s11047-020-09820-4 10.1177/003754977101700504 10.1111/itor.12001 10.3390/app11115053 10.3390/math10050800 10.1016/j.ejco.2021.100012 10.1007/s10107-006-0006-3 10.1137/1.9781611972672 10.1109/TAC.1968.1098903 10.1145/3479242.3487323 10.1145/3121050.3121108 10.1016/j.eswa.2022.116696 |
| ContentType | Journal Article |
| Copyright | The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. |
| Copyright_xml | – notice: The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. |
| DBID | AAYXX CITATION |
| DOI | 10.1007/s11590-023-02038-0 |
| DatabaseName | CrossRef |
| DatabaseTitle | CrossRef |
| DatabaseTitleList | |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Engineering Mathematics |
| EISSN | 1862-4480 |
| EndPage | 2111 |
| ExternalDocumentID | 10_1007_s11590_023_02038_0 |
| GroupedDBID | -5D -5G -BR -EM -Y2 -~C .VR 06D 0R~ 0VY 123 1N0 203 2J2 2JN 2JY 2KG 2KM 2LR 2VQ 2~H 30V 4.4 406 408 409 40D 40E 5VS 67Z 6NX 8TC 8UJ 95- 95. 95~ 96X AAAVM AABHQ AACDK AAHNG AAIAL AAJBT AAJKR AANZL AARHV AARTL AASML AATNV AATVU AAUYE AAWCG AAYIU AAYQN AAYTO AAYZH ABAKF ABBXA ABDZT ABECU ABFTV ABHQN ABJNI ABJOX ABKCH ABMNI ABMQK ABNWP ABQBU ABSXP ABTEG ABTHY ABTKH ABTMW ABULA ABWNU ABXPI ACAOD ACBXY ACDTI ACGFS ACHSB ACHXU ACKNC ACMDZ ACMLO ACOKC ACOMO ACPIV ACSNA ACZOJ ADHHG ADHIR ADINQ ADKNI ADKPE ADRFC ADTPH ADURQ ADYFF ADZKW AEBTG AEFQL AEGAL AEGNC AEJHL AEJRE AEMSY AENEX AEOHA AEPYU AESKC AETLH AEVLU AEXYK AFBBN AFGCZ AFLOW AFQWF AFWTZ AFZKB AGAYW AGDGC AGJBK AGMZJ AGQEE AGQMX AGRTI AGWIL AGWZB AGYKE AHAVH AHBYD AHKAY AHSBF AHYZX AIAKS AIGIU AIIXL AILAN AITGF AJBLW AJRNO AJZVZ ALMA_UNASSIGNED_HOLDINGS ALWAN AMKLP AMXSW AMYLF AMYQR AOCGG ARMRJ AXYYD AYJHY B-. BA0 BAPOH BDATZ BGNMA BSONS CAG COF CS3 CSCUP DDRTE DNIVK DPUIP DU5 EBLON EBS EIOEI EJD ESBYG FERAY FFXSO FIGPU FINBP FNLPD FRRFC FSGXE FWDCC GGCAI GGRSB GJIRD GNWQR GQ6 GQ7 GQ8 GXS H13 HF~ HG5 HG6 HLICF HMJXF HQYDN HRMNR HZ~ IJ- IKXTQ IWAJR IXC IXD IXE IZIGR IZQ I~X I~Z J-C J0Z J9A JBSCW JCJTX JZLTJ KDC KOV LLZTM M4Y MA- N9A NPVJJ NQJWS NU0 O9- O93 O9J OAM P2P P9M PF0 PT4 QOS R89 R9I ROL RPX RSV S16 S1Z S27 S3B SAP SDH SHX SISQX SJYHP SMT SNE SNPRN SNX SOHCF SOJ SPISZ SRMVM SSLCW STPWE SZN T13 TSG TSK TSV TUC U2A UG4 UOJIU UTJUX UZXMN VC2 VFIZW W23 W48 WK8 YLTOR Z45 Z7R Z7Y Z83 Z88 ZMTXR ~A9 AAPKM AAYXX ABBRH ABDBE ABFSG ABRTQ ACSTC AEZWR AFDZB AFHIU AFOHR AHPBZ AHWEU AIXLP ATHPR AYFIA CITATION |
| ID | FETCH-LOGICAL-c291t-ccecacb09e5e2dfa6f875a7fc20652e7c8dbbf7939f47eaa527340a07fe39cc03 |
| IEDL.DBID | AGYKE |
| ISSN | 1862-4472 |
| IngestDate | Thu Apr 24 23:05:49 EDT 2025 Wed Oct 01 02:12:05 EDT 2025 Fri Feb 21 02:37:19 EST 2025 |
| IsPeerReviewed | true |
| IsScholarly | true |
| Issue | 9 |
| Keywords | Global optimization Metaheuristics Multi-armed bandit Reinforcement learning Random search |
| Language | English |
| LinkModel | DirectLink |
| MergedId | FETCHMERGED-LOGICAL-c291t-ccecacb09e5e2dfa6f875a7fc20652e7c8dbbf7939f47eaa527340a07fe39cc03 |
| ORCID | 0000-0001-8828-2919 0000-0002-1829-9587 |
| PageCount | 21 |
| ParticipantIDs | crossref_citationtrail_10_1007_s11590_023_02038_0 crossref_primary_10_1007_s11590_023_02038_0 springer_journals_10_1007_s11590_023_02038_0 |
| ProviderPackageCode | CITATION AAYXX |
| PublicationCentury | 2000 |
| PublicationDate | 20241200 2024-12-00 |
| PublicationDateYYYYMMDD | 2024-12-01 |
| PublicationDate_xml | – month: 12 year: 2024 text: 20241200 |
| PublicationDecade | 2020 |
| PublicationPlace | Berlin/Heidelberg |
| PublicationPlace_xml | – name: Berlin/Heidelberg |
| PublicationTitle | Optimization letters |
| PublicationTitleAbbrev | Optim Lett |
| PublicationYear | 2024 |
| Publisher | Springer Berlin Heidelberg |
| Publisher_xml | – name: Springer Berlin Heidelberg |
| References | Matyas (CR5) 1965; 26 CR19 Plevris, Bakas, Solorzano (CR10) 2021; 11 CR18 CR17 CR15 CR13 Sörensen (CR1) 2015; 22 Sutton, Barto (CR12) 2018 White (CR6) 1971; 17 Hameed, Rajagopalan (CR11) 2022; 10 Rastrigin (CR4) 1963; 24 Stork, Eiben, Bartz-Beielstein (CR2) 2020; 21 CR3 CR8 CR7 Locatelli, Schoen (CR9) 2021; 9 CR24 CR23 CR22 CR21 Grosso, Locatelli, Schoen (CR25) 2007; 110 Mohamed, Hadi, Mohamed (CR20) 2020; 11 Casella, Berger (CR14) 2001 Jamil, Yang (CR16) 2013; 4 Vagelis Plevris (2038_CR10) 2021; 11 Kenneth Sörensen (2038_CR1) 2015; 22 A Grosso (2038_CR25) 2007; 110 2038_CR24 2038_CR23 2038_CR22 2038_CR21 RC White Jr (2038_CR6) 1971; 17 2038_CR7 AW Mohamed (2038_CR20) 2020; 11 2038_CR8 2038_CR3 2038_CR17 J Matyas (2038_CR5) 1965; 26 2038_CR19 2038_CR18 J Stork (2038_CR2) 2020; 21 RS Sutton (2038_CR12) 2018 2038_CR13 2038_CR15 M Jamil (2038_CR16) 2013; 4 ASSS Hameed (2038_CR11) 2022; 10 Marco Locatelli (2038_CR9) 2021; 9 LA Rastrigin (2038_CR4) 1963; 24 G Casella (2038_CR14) 2001 |
| References_xml | – ident: CR22 – ident: CR18 – volume: 11 start-page: 1501 issue: 7 year: 2020 end-page: 1529 ident: CR20 article-title: Gaining-sharing knowledge based algorithm for solving optimization problems: a novel nature-inspired algorithm publication-title: Int. J. Mach. Learn. Cybernet. doi: 10.1007/s13042-019-01053-x – volume: 4 start-page: 150 year: 2013 end-page: 194 ident: CR16 article-title: A literature survey of benchmark functions for global optimisation problems publication-title: Int. J. Math. Model. Numer. Optim. – volume: 21 start-page: 219 year: 2020 end-page: 242 ident: CR2 article-title: A new taxonomy of global optimization algorithms publication-title: Nat. Comput. doi: 10.1007/s11047-020-09820-4 – volume: 17 start-page: 197 issue: 5 year: 1971 end-page: 205 ident: CR6 article-title: A survey of random methods for parameter optimization publication-title: Simulation doi: 10.1177/003754977101700504 – volume: 22 start-page: 3 issue: 1 year: 2015 end-page: 18 ident: CR1 article-title: Metaheuristics-the metaphor exposed publication-title: Int. Trans. Oper. Res. doi: 10.1111/itor.12001 – ident: CR8 – volume: 26 start-page: 246 issue: 2 year: 1965 end-page: 253 ident: CR5 article-title: Random optimization publication-title: Automat. Remote control – volume: 11 start-page: 5053 issue: 11 year: 2021 ident: CR10 article-title: Pure random orthogonal search (PROS): a plain and elegant parameterless algorithm for global optimization publication-title: Appl. Sci. doi: 10.3390/app11115053 – volume: 24 start-page: 1337 year: 1963 end-page: 1342 ident: CR4 article-title: The convergence of the random search method in the extremal control of a many parameter system publication-title: Automat. Remote Control – ident: CR23 – volume: 10 start-page: 800 issue: 5 year: 2022 ident: CR11 article-title: SPGD: search party gradient descent algorithm, a simple gradient-based parallel algorithm for bound-constrained optimization publication-title: Mathematics doi: 10.3390/math10050800 – ident: CR21 – ident: CR19 – volume: 9 year: 2021 ident: CR9 article-title: (Global) Optimization: historical notes and recent developments publication-title: EURO J. Comput. Optim. doi: 10.1016/j.ejco.2021.100012 – ident: CR3 – ident: CR15 – year: 2001 ident: CR14 publication-title: Statistical Inference – ident: CR17 – ident: CR13 – year: 2018 ident: CR12 publication-title: Reinforcement learning: an introduction – volume: 110 start-page: 373 year: 2007 end-page: 404 ident: CR25 article-title: A population-based approach for hard global optimization problems based on dissimilarity measures publication-title: Math. Progr. doi: 10.1007/s10107-006-0006-3 – ident: CR7 – ident: CR24 – ident: 2038_CR8 – ident: 2038_CR24 – volume: 4 start-page: 150 year: 2013 ident: 2038_CR16 publication-title: Int. J. Math. Model. Numer. Optim. – volume: 17 start-page: 197 issue: 5 year: 1971 ident: 2038_CR6 publication-title: Simulation doi: 10.1177/003754977101700504 – volume: 22 start-page: 3 issue: 1 year: 2015 ident: 2038_CR1 publication-title: Int. Trans. Oper. Res. doi: 10.1111/itor.12001 – volume: 110 start-page: 373 year: 2007 ident: 2038_CR25 publication-title: Math. Progr. doi: 10.1007/s10107-006-0006-3 – ident: 2038_CR13 – volume: 26 start-page: 246 issue: 2 year: 1965 ident: 2038_CR5 publication-title: Automat. Remote control – ident: 2038_CR3 doi: 10.1137/1.9781611972672 – volume-title: Statistical Inference year: 2001 ident: 2038_CR14 – ident: 2038_CR17 – ident: 2038_CR15 – volume: 11 start-page: 1501 issue: 7 year: 2020 ident: 2038_CR20 publication-title: Int. J. Mach. Learn. Cybernet. doi: 10.1007/s13042-019-01053-x – volume: 10 start-page: 800 issue: 5 year: 2022 ident: 2038_CR11 publication-title: Mathematics doi: 10.3390/math10050800 – ident: 2038_CR23 – ident: 2038_CR7 doi: 10.1109/TAC.1968.1098903 – ident: 2038_CR21 doi: 10.1145/3479242.3487323 – volume-title: Reinforcement learning: an introduction year: 2018 ident: 2038_CR12 – volume: 9 year: 2021 ident: 2038_CR9 publication-title: EURO J. Comput. Optim. doi: 10.1016/j.ejco.2021.100012 – volume: 21 start-page: 219 year: 2020 ident: 2038_CR2 publication-title: Nat. Comput. doi: 10.1007/s11047-020-09820-4 – volume: 24 start-page: 1337 year: 1963 ident: 2038_CR4 publication-title: Automat. Remote Control – ident: 2038_CR22 doi: 10.1145/3121050.3121108 – ident: 2038_CR18 – ident: 2038_CR19 doi: 10.1016/j.eswa.2022.116696 – volume: 11 start-page: 5053 issue: 11 year: 2021 ident: 2038_CR10 publication-title: Appl. Sci. doi: 10.3390/app11115053 |
| SSID | ssj0061199 |
| Score | 2.357343 |
| Snippet | We live in a world where waves of novel nature-inspired metaheuristic algorithms keep hitting the shore repeatedly. This never-ending surge of new... |
| SourceID | crossref springer |
| SourceType | Enrichment Source Index Database Publisher |
| StartPage | 2091 |
| SubjectTerms | Computational Intelligence Mathematics Mathematics and Statistics Numerical and Computational Physics Operations Research/Decision Theory Optimization Original Paper Simulation |
| Title | NPROS: A Not So Pure Random Orthogonal search algorithm—A suite of random optimization algorithms driven by reinforcement learning |
| URI | https://link.springer.com/article/10.1007/s11590-023-02038-0 |
| Volume | 18 |
| hasFullText | 1 |
| inHoldings | 1 |
| isFullTextHit | |
| isPrint | |
| journalDatabaseRights | – providerCode: PRVLSH databaseName: SpringerLink Journals customDbUrl: mediaType: online eissn: 1862-4480 dateEnd: 99991231 omitProxy: false ssIdentifier: ssj0061199 issn: 1862-4472 databaseCode: AFBBN dateStart: 20070101 isFulltext: true providerName: Library Specific Holdings – providerCode: PRVAVX databaseName: SpringerLINK - Czech Republic Consortium customDbUrl: eissn: 1862-4480 dateEnd: 99991231 omitProxy: false ssIdentifier: ssj0061199 issn: 1862-4472 databaseCode: AGYKE dateStart: 20070101 isFulltext: true titleUrlDefault: http://link.springer.com providerName: Springer Nature – providerCode: PRVAVX databaseName: SpringerLink Journals (ICM) customDbUrl: eissn: 1862-4480 dateEnd: 99991231 omitProxy: true ssIdentifier: ssj0061199 issn: 1862-4472 databaseCode: U2A dateStart: 20070101 isFulltext: true titleUrlDefault: http://www.springerlink.com/journals/ providerName: Springer Nature |
| link | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwlV1LT9tAEF6VcIFDS1tQoSWaQ2-t0Xrj-MHNoKRRKwKCRqIna5-hKomR7RzgxKE_ob-wv4RZe00KqpA4Whpbq9ndeXjm-4aQjwE1MRd4v5VSmKCgwfOEHwuPWXSO4XE_phbvfDQOR5Pg63n_3IHCyrbbvS1J1pZ6CXZDz0s99DGerZ7FHibqqzXfVoespl9-fBu0Fjj0m7mRfmwRQUHEHFjm_1956JAeVkNrJzN8RSbt8prekl97i0rsyZtHzI3PXf8GeemiTkibY_KavNDzN2T9Hy5CfDq6J3At35Lf45PT47N9SGGcV3CWw8mi0HDK5yqfwXFRXeRTG8JDc1GAX07z4md1Mft7-yeFcoFxLOQGikY-R7M0c3jPpWgJqrCmFsQ1FLomcJX1v0pwkyymm2QyHHw_HHluYIMnWeJXnpRaciloovuaKcNDg9kQj4xkGOgwHclYCWHQIiQmiDTnlvwtoJxGRvcSKWlvi3Tm-Vy_I5AIRpWOMb_iLNCYxojQUKlFQmWoqDTbxG93LZOOzdwO1bjMljzMVt8Z6jur9Z3RbfLp_p2rhsvjSenP7T5m7l6XT4jvPE_8PVljGB41jTEfSKcqFnoXw5tKdPE0Dw8Oxl13qrtkZcLSO4KV9LU |
| linkProvider | Springer Nature |
| linkToHtml | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwlV25TsNAEF0hKIACcYqbKejA0npxfNBFiCgcCQiIRGftmSCRGNlOQUfBJ_CFfAmzPhKQEBKlpfEWnp2ZN959bwg59KgJucD4Vkphg4IJzxFuKBxm2TmGh42QWr5zp-u3e97lY-OxIoVl9W33-kiyyNRTshtWXupgjXHs6VnoYKM-ZwWsrGJ-jzXr_Ou75dRIN7R8IC9gFVXm9zV-lqOfZ6FFiWktk6UKG0KzdOYKmdGjVbL4TTEQnzoTmdVsjbx3b-9u7k-hCd0kh_sEbsephjs-UskQbtJ8kPQt0IZyOwN_7ifpUz4Yfr59NCEbI9qExEBa2ieYPIYVK3NqmoFKbUIE8QqpLmRWZfFHEap5E_110mudP5y1nWqsgiNZ5OaOlFpyKWikG5opw32DPQsPjGQIR5gOZKiEMBi3kfECzbmVaPMop4HRJ5GU9GSDzI6Skd4kEAlGlQ6xC-LM09hsCN9QqUVEpa-oNFvErb9uLCvNcTv64jmeqiVbj8TokbjwSEy3yNHknZdSceNP6-PaaXEVfdkf5tv_Mz8g8-2HznV8fdG92iELDAFNeZVll8zm6VjvISDJxX6x_74AoX_YRQ |
| linkToPdf | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwlV1JS8NAFB5EQfQgrrj7Dt40OJmmWbwVtdStFrXgLcxaBZtImh68efAn-Av9Jb5J0lZBBI-Blznk7XnzfY-QfY-akAv0b6UUNigY8BzhhsJhFp1jeFgPqcU7X7f9Vte7eKg_fEPxF7fdRyPJEtNgWZqS_OhFmaMJ8A2zMHUw3zh2khY62LTPeJYoAS26yxqjWOy75QZJN7TYIC9gFWzm9zN-pqafc9Ei3TQXyUJVJ0KjVOwSmdLJMpn_xh6IT9djytXBCnlvd25v7o6hAe00h7sUOsNMwy1PVNqHmyx_THu26IbStIE_99LsKX_sf759NGAwxMoTUgNZKZ9iIOlXCM2J6ABUZoMjiFfIdEG5Kou_i1Dtnuitkm7z7P6k5VQrFhzJIjd3pNSSS0EjXddMGe4b7F94YCTD0oTpQIZKCIM-HBkv0JxbujaPchoYXYukpLU1Mp2kiV4nEAlGlQ6xI-LM09h4CN9QqUVEpa-oNBvEHX3dWFb843YNxnM8YU62GolRI3GhkZhukIPxOy8l-8af0ocjpcWVJw7-EN_8n_geme2cNuOr8_blFpljWNuUt1q2yXSeDfUO1ia52C3M7wsfHNyB |
| openUrl | ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=NPROS%3A+A+Not+So+Pure+Random+Orthogonal+search+algorithm%E2%80%94A+suite+of+random+optimization+algorithms+driven+by+reinforcement+learning&rft.jtitle=Optimization+letters&rft.au=Hameed%2C+A.+S.+Syed+Shahul&rft.au=Rajagopalan%2C+Narendran&rft.date=2024-12-01&rft.pub=Springer+Berlin+Heidelberg&rft.issn=1862-4472&rft.eissn=1862-4480&rft.volume=18&rft.issue=9&rft.spage=2091&rft.epage=2111&rft_id=info:doi/10.1007%2Fs11590-023-02038-0&rft.externalDocID=10_1007_s11590_023_02038_0 |
| thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=1862-4472&client=summon |
| thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=1862-4472&client=summon |
| thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=1862-4472&client=summon |