{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,20]],"date-time":"2026-02-20T03:03:26Z","timestamp":1771556606527,"version":"3.50.1"},"reference-count":44,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2023,6,7]],"date-time":"2023-06-07T00:00:00Z","timestamp":1686096000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Ministry of Science and Higher Education of the Russian Federation","award":["075-15-2022-1121"],"award-info":[{"award-number":["075-15-2022-1121"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The impact of neutrosophy has increased rapidly in many areas of science and technology in recent years. Furthermore, numerous applications of the neutrosophic theory have become more usual. We aim to use neutrosophy to enhance Dai\u2013Liao conjugate gradient (CG) iterative method. In particular, we suggest and explore a new neutrosophic logic system intended to compute the essential parameter t required in Dai\u2013Liao CG iterations. Theoretical examination and numerical experiments signify the effectiveness of the introduced method for controlling t. By incorporation of the neutrosophy in the Dai\u2013Liao conjugate gradient principle, we established novel Dai\u2013Liao CG iterations for solving large-scale unconstrained optimization problems. Global convergence is proved under standard assumptions and with the use of the inexact line search. Finally, computational evidence shows the computational effectiveness of the proposed fuzzy neutrosophic Dai\u2013Liao CG method.<\/jats:p>","DOI":"10.3390\/sym15061217","type":"journal-article","created":{"date-parts":[[2023,6,8]],"date-time":"2023-06-08T01:34:32Z","timestamp":1686188072000},"page":"1217","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Fuzzy Adaptive Parameter in the Dai\u2013Liao Optimization Method Based on Neutrosophy"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0655-3741","authenticated-orcid":false,"given":"Predrag S.","family":"Stanimirovi\u0107","sequence":"first","affiliation":[{"name":"Faculty of Sciences and Mathematics, University of Ni\u0161, Vi\u0161egradska 33, 18000 Ni\u0161, Serbia"},{"name":"Laboratory \u201cHybrid Methods of Modelling and Optimization in Complex Systems\u201d, Siberian Federal University, Prosp. Svobodny 79, 660041 Krasnoyarsk, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9179-0965","authenticated-orcid":false,"given":"Branislav D.","family":"Ivanov","sequence":"additional","affiliation":[{"name":"Faculty of Sciences and Mathematics, University of Ni\u0161, Vi\u0161egradska 33, 18000 Ni\u0161, Serbia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6846-3074","authenticated-orcid":false,"given":"Dragi\u0161a","family":"Stanujki\u0107","sequence":"additional","affiliation":[{"name":"Technical Faculty in Bor, University of Belgrade, Vojske Jugoslavije 12, 19210 Bor, Serbia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0667-4001","authenticated-orcid":false,"given":"Lev A.","family":"Kazakovtsev","sequence":"additional","affiliation":[{"name":"Laboratory \u201cHybrid Methods of Modelling and Optimization in Complex Systems\u201d, Siberian Federal University, Prosp. Svobodny 79, 660041 Krasnoyarsk, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8704-1670","authenticated-orcid":false,"given":"Vladimir N.","family":"Krutikov","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, Kemerovo State University, 6 Krasnaya Street, 650043 Kemerovo, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5308-2503","authenticated-orcid":false,"given":"Darjan","family":"Karaba\u0161evi\u0107","sequence":"additional","affiliation":[{"name":"Faculty of Applied Management, Economics and Finance, University Business Academy in Novi Sad, Jevrejska 24, 11000 Belgrade, Serbia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,6,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1007\/s002450010019","article-title":"New conjugacy conditions and related nonlinear conjugate gradient methods","volume":"43","author":"Dai","year":"2001","journal-title":"Appl. Math. Optim."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"577","DOI":"10.1080\/10556788.2015.1124431","article-title":"A sufficient descent conjugate gradient method and its global convergence","volume":"31","author":"Cheng","year":"2016","journal-title":"Optim. Methods Softw."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"435495","DOI":"10.5402\/2012\/435495","article-title":"A descent Dai-Liao conjugate gradient method based on a modified secant equation and its global convergence","volume":"2012","author":"Livieris","year":"2012","journal-title":"Isrn Comput. Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"843","DOI":"10.1080\/10556788.2014.1001511","article-title":"A new class of efficient and globally convergent conjugate gradient methods in the Dai-Liao family","volume":"30","author":"Peyghami","year":"2015","journal-title":"Optim. Methods Softw."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1023\/B:COAP.0000026885.81997.88","article-title":"Global convergence properties of nonlinear conjugate gradient methods with modified secant condition","volume":"28","author":"Yabe","year":"2004","journal-title":"Comput. Optim. Appl."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"279891","DOI":"10.1155\/2014\/279891","article-title":"A hybrid of DL and WYL nonlinear conjugate gradient methods","volume":"2014","author":"Yao","year":"2014","journal-title":"Abstr. Appl. Anal."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"730454","DOI":"10.1155\/2013\/730454","article-title":"A conjugate gradient method with global convergence for large-scale unconstrained optimization problems","volume":"2013","author":"Yao","year":"2013","journal-title":"J. Appl. Math."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"502","DOI":"10.1007\/s10957-017-1140-1","article-title":"Two new Dai-Liao-type conjugate gradient methods for unconstrained optimization problems","volume":"175","author":"Zheng","year":"2017","journal-title":"J. Optim. Theory Appl."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"707","DOI":"10.1080\/10556780500137041","article-title":"A nonlinear conjugate gradient method based on the MBFGS secant condition","volume":"21","author":"Zhou","year":"2006","journal-title":"Optim. Methods Softw."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"443","DOI":"10.1007\/s40065-019-0264-6","article-title":"A Dai\u2013Liao conjugate gradient method via modified secant equation for system of nonlinear equations","volume":"9","author":"Waziri","year":"2020","journal-title":"Arab. J. Math."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Khoshsimaye\u2013Bargard, M., and Ashrafi, A. (2023). A descent spectral Dai\u2013Liao method based on the quasi\u2013Newton aspects. Numer. Algor.","DOI":"10.1007\/s11075-023-01506-z"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1051\/ro\/2022213","article-title":"A survey on the Dai-Liao family of nonlinear conjugate gradient methods","volume":"57","year":"2023","journal-title":"RAIRO-Oper. Res."},{"key":"ref_13","first-page":"319","article-title":"Open problems in nonlinear conjugate gradient algorithms for unconstrained optimization","volume":"34","author":"Andrei","year":"2011","journal-title":"Bull. Malays. Math. Sci. Soc."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"170","DOI":"10.1137\/030601880","article-title":"A new conjugate gradient method with guaranteed descent and an efficient line search","volume":"16","author":"Hager","year":"2005","journal-title":"SIAM J. Optim."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1145\/1132973.1132979","article-title":"Algorithm 851: CG DESCENT, a conjugate gradient method with guaranteed descent","volume":"32","author":"Hager","year":"2006","journal-title":"Acm Trans. Math. Softw."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"296","DOI":"10.1137\/100813026","article-title":"A nonlinear conjugate gradient algorithm with an optimal property and an improved wolfe line search","volume":"23","author":"Dai","year":"2013","journal-title":"SIAM J. Optim."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"625","DOI":"10.1016\/j.ejor.2013.11.012","article-title":"The Dai-Liao nonlinear conjugate gradient method with optimal parameter choices","volume":"234","author":"Ghanbari","year":"2014","journal-title":"Europ. J. Oper. Res."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1273","DOI":"10.1007\/s11075-017-0362-5","article-title":"A Dai-Liao conjugate gradient algorithm with clustering of eigenvalues","volume":"77","author":"Andrei","year":"2018","journal-title":"Numer. Algor."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"285","DOI":"10.1007\/s10288-014-0255-6","article-title":"On the sufficient descent condition of the Hager-Zhang conjugate gradient methods","volume":"12","year":"2014","journal-title":"4OR-Q J. Oper. Res."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"112708","DOI":"10.1016\/j.cam.2019.112708","article-title":"An efficient Dai\u2013Liao type conjugate gradient method by reformulating the CG parameter in the search direction equation","volume":"371","author":"Lotfi","year":"2020","journal-title":"J. Comput. Appl. Math."},{"key":"ref_21","first-page":"6693401","article-title":"A Novel Value for the Parameter in the Dai-Liao-Type Conjugate Gradient Method","volume":"2021","author":"Ivanov","year":"2021","journal-title":"J. Funct. Spaces"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"338","DOI":"10.1016\/S0019-9958(65)90241-X","article-title":"Fuzzy sets","volume":"8","author":"Zadeh","year":"1965","journal-title":"Inf. Control"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1016\/S0165-0114(86)80034-3","article-title":"Intuitionistic fuzzy sets","volume":"20","author":"Atanassov","year":"1986","journal-title":"Fuzzy Sets Syst."},{"key":"ref_24","unstructured":"Smarandache, F. (1999). A Unifying Field in Logics, Neutrosophy: Neutrosophic Probability, Set and Logic, American Research Press."},{"key":"ref_25","first-page":"410","article-title":"Single valued neutrosophic sets","volume":"4","author":"Wang","year":"2010","journal-title":"Multispace Multistruct."},{"key":"ref_26","unstructured":"Smarandache, F. (2023, April 30). Special Issue \u201cNew types of Neutrosophic Set\/Logic\/Probability, Neutrosophic Over-\/Under-\/Off-Set, Neutrosophic Refined Set, and their Extension to Plithogenic Set\/Logic\/Probability, with Applications\u201d. Symmetry. Available online: https:\/\/www.mdpi.com\/journal\/symmetry\/special_issues\/Neutrosophic_Set_Logic_Probability."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Mishra, K., Kandasamy, I., Kandasamy, W.B., and Smarandache, F. (2020). A novel framework using neutrosophy for integrated speech and text sentiment analysis. Symmetry, 12.","DOI":"10.3390\/sym12101715"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Tu, A., Ye, J., and Wang, B. (2018). Symmetry measures of simplified neutrosophic sets for multiple attribute decision-making problems. Symmetry, 10.","DOI":"10.3390\/sym10050144"},{"key":"ref_29","unstructured":"Smarandache, F. (2022). Collected Papers (On Physics, Artificial Intelligence, Health Issues, Decision Making, Economics, Statistics), Global Knowledge Publishing House. Available online: http:\/\/fs.unm.edu\/CP11.pdf."},{"key":"ref_30","unstructured":"Smarandache, F. (2022). Collected Papers (On Various Scientific Topics), Global Knowledge Publishing House. Available online: http:\/\/fs.unm.edu\/CP13.pdf."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Stanimirovi\u0107, P.S., Ivanov, B., Stanujki\u0107, D., Katsikis, V.N., Mourtas, S.D., Kazakovtsev, L.A., and Edalatpanah, S.A. (2023). Improvement of Unconstrained Optimization Methods Based on Symmetry Involved in Neutrosophy. Symmetry, 15.","DOI":"10.3390\/sym15010250"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"2434","DOI":"10.1109\/TII.2021.3093115","article-title":"Design and analysis of a hybrid GNN-ZNN model with a fuzzy adaptive factor for matrix inversion","volume":"18","author":"Dai","year":"2022","journal-title":"IEEE Trans. Ind. Inform."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1006","DOI":"10.1109\/TFUZZ.2016.2574915","article-title":"A hierarchical fused fuzzy deep neural network for data classification","volume":"25","author":"Deng","year":"2017","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"2603","DOI":"10.1109\/TFUZZ.2020.3005272","article-title":"A novel fuzzy-power zeroing neural network model for time-variant matrix Moore-Penrose inversion with guaranteed performance","volume":"29","author":"Jia","year":"2021","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1544","DOI":"10.1109\/TFUZZ.2020.2981001","article-title":"Design and Application of an Adaptive Fuzzy Control Strategy to Zeroing Neural Network for Solving Time-Variant QP Problem","volume":"29","author":"Jia","year":"2021","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"3426","DOI":"10.1109\/TFUZZ.2021.3115969","article-title":"Zeroing neural network with fuzzy parameter for computing pseudoinverse of arbitrary matrix","volume":"30","author":"Katsikis","year":"2022","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"409","DOI":"10.6028\/jres.049.044","article-title":"Methods of conjugate gradients for solving linear systems","volume":"49","author":"Hestenes","year":"1952","journal-title":"J. Res. Nat. Bur. Stand."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1007\/s11075-006-9023-9","article-title":"An acceleration of gradient descent algorithm with backtracking for unconstrained optimization","volume":"42","author":"Andrei","year":"2006","journal-title":"Numer. Algorithms"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"503","DOI":"10.1007\/s11075-009-9350-8","article-title":"Accelerated gradient descent methods with line search","volume":"54","year":"2010","journal-title":"Numer. Algorithms"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"1217","DOI":"10.1080\/01630560701749524","article-title":"A two-term PRP-based descent method","volume":"28","author":"Cheng","year":"2007","journal-title":"Numer. Funct. Anal. Optim."},{"key":"ref_41","unstructured":"Abadie, J. (1970). Integer and Nonlinear Programming, Springer."},{"key":"ref_42","first-page":"147","article-title":"An unconstrained optimization test functions collection","volume":"10","author":"Andrei","year":"2008","journal-title":"Adv. Model. Optim."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1145\/200979.201043","article-title":"CUTE: Constrained and unconstrained testing environments","volume":"21","author":"Bongartz","year":"1995","journal-title":"ACM Trans. Math. Softw."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1007\/s101070100263","article-title":"Benchmarking optimization software with performance profiles","volume":"91","author":"Dolan","year":"2002","journal-title":"Math. Program."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/6\/1217\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:50:14Z","timestamp":1760125814000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/6\/1217"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,6,7]]},"references-count":44,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2023,6]]}},"alternative-id":["sym15061217"],"URL":"https:\/\/doi.org\/10.3390\/sym15061217","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,6,7]]}}}