{"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":1771556606496,"version":"3.50.1"},"reference-count":47,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2021,7,28]],"date-time":"2021-07-28T00:00:00Z","timestamp":1627430400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>The conjugate gradient method is one of the most popular methods to solve large-scale unconstrained optimization problems since it does not require the second derivative, such as Newton\u2019s method or approximations. Moreover, the conjugate gradient method can be applied in many fields such as neural networks, image restoration, etc. Many complicated methods are proposed to solve these optimization functions in two or three terms. In this paper, we propose a simple, easy, efficient, and robust conjugate gradient method. The new method is constructed based on the Liu and Storey method to overcome the convergence problem and descent property. The new modified method satisfies the convergence properties and the sufficient descent condition under some assumptions. The numerical results show that the new method outperforms famous CG methods such as CG-Descent 5.3, Liu and Storey, and Dai and Liao. The numerical results include the number of iterations and CPU time.<\/jats:p>","DOI":"10.3390\/a14080227","type":"journal-article","created":{"date-parts":[[2021,7,28]],"date-time":"2021-07-28T11:12:52Z","timestamp":1627470772000},"page":"227","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["A Modified Liu and Storey Conjugate Gradient Method for Large Scale Unconstrained Optimization Problems"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5877-9051","authenticated-orcid":false,"given":"Zabidin","family":"Salleh","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Ocean Engineering Technology and Informatics, Universiti Malaysia Terengganu, Kuala Nerus 21030, Terengganu, Malaysia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ghaliah","family":"Alhamzi","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 13318, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ibitsam","family":"Masmali","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, Jazan University, Jazan 45142, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8283-0432","authenticated-orcid":false,"given":"Ahmad","family":"Alhawarat","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Ocean Engineering Technology and Informatics, Universiti Malaysia Terengganu, Kuala Nerus 21030, Terengganu, Malaysia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,7,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"226","DOI":"10.1137\/1011036","article-title":"Convergence Conditions for Ascent Methods","volume":"11","author":"Wolfe","year":"1969","journal-title":"SIAM Rev."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1137\/1013035","article-title":"Convergence Conditions for Ascent Methods. II: Some Corrections","volume":"13","author":"Wolfe","year":"1971","journal-title":"SIAM Rev."},{"key":"ref_3","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. Natl. Inst. Stand. Technol."},{"key":"ref_4","first-page":"35","article-title":"Note sur la convergence de m\u00e9thodes de directions conjugu\u00e9es","volume":"3. R1","author":"Elijah","year":"1969","journal-title":"ESAIM: Math. Model. Numer. Anal. Mod\u00e9lisation Math\u00e9matique Et Anal. Num\u00e9rique"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1007\/BF00940464","article-title":"Efficient generalized conjugate gradient algorithms, part 1: Theory","volume":"69","author":"Liu","year":"1991","journal-title":"J. Optim. Theory Appl."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1093\/comjnl\/7.2.149","article-title":"Function minimization by conjugate gradients","volume":"7","author":"Fletcher","year":"1964","journal-title":"Comput. J."},{"key":"ref_7","unstructured":"Fletcher, R. (1987). Practical Methods of Optimization, Unconstrained Optimization, Wiley."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1137\/S1052623497318992","article-title":"A non-linear conjugate gradient method with a strong global convergence property","volume":"10","author":"Dai","year":"1999","journal-title":"SIAM J. Optim."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"122","DOI":"10.1007\/BFb0099521","article-title":"Non-convex minimization calculations and the conjugate gradient method, Numerical Analysis (Dundee, 1983)","volume":"Volume 1066","author":"Powell","year":"1984","journal-title":"Lecture Notes in Mathematics"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1137\/0802003","article-title":"Global convergence properties of conjugate gradient methods for optimization","volume":"2","author":"Gilbert","year":"1992","journal-title":"Siam. J. Optim."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1093\/imanum\/5.1.121","article-title":"Descent property and global convergence of the Fletcher-Reeves method with inexact line search","volume":"5","year":"1985","journal-title":"IMA J. Numer. Anal."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1007\/s002450010019","article-title":"New conjugacy conditions and related non-linear conjugate gradient methods","volume":"43","author":"Dai","year":"2001","journal-title":"Appl. Math. Optim."},{"key":"ref_13","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_14","doi-asserted-by":"crossref","first-page":"2150","DOI":"10.1137\/120898097","article-title":"The limited memory conjugate gradient method","volume":"23","author":"Hager","year":"2013","journal-title":"Siam. J. Optim."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1023\/A:1021898630001","article-title":"New quasi-Newton equation and related methods for unconstrained optimization","volume":"102","author":"Zhang","year":"1999","journal-title":"J. Optim. Theory Appl."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1016\/S0377-0427(00)00713-5","article-title":"Properties and numerical performance of quasi-Newton methods with modified quasi-Newton equation","volume":"137","author":"Zhang","year":"2001","journal-title":"J. Comput. Appl. Math."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1023\/B:COAP.0000026885.81997.88","article-title":"Global convergence properties of non-linear conjugate gradient methods with modified secant relation","volume":"28","author":"Yabe","year":"2004","journal-title":"Comput. Optim. Appl."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"621","DOI":"10.1080\/01630563.2019.1669641","article-title":"A Conjugate Gradient Method Based on a Modified Secant Relation for Unconstrained Optimization","volume":"41","author":"Dehghani","year":"2020","journal-title":"Numer. Funct. Anal. Optim."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s40314-021-01557-9","article-title":"An improved Polak\u2013Ribi\u00e8re\u2013Polyak conjugate gradient method with an efficient restart direction","volume":"40","author":"Jiang","year":"2021","journal-title":"Comput. Appl. Math."},{"key":"ref_20","first-page":"1341","article-title":"The convergence properties of some new conjugate gradient methods","volume":"183","author":"Wei","year":"2006","journal-title":"Appl. Math. Comput."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"177v188","DOI":"10.1080\/10556788.2012.755182","article-title":"A modified Polak\u2013Ribi\u2019e re\u2013Polyak descent method for unconstrained optimization","volume":"29","author":"Qu","year":"2014","journal-title":"Optim. Methods Softw."},{"key":"ref_22","first-page":"381","article-title":"A note about WYL\u2019s conjugate gradient method and its applications","volume":"191","author":"Shengwei","year":"2007","journal-title":"Appl. Math. Comput."},{"key":"ref_23","first-page":"2269","article-title":"An improved Wei\u2013Yao\u2013Liu non-linear conjugate gradient method for optimization computation","volume":"215","author":"Zhang","year":"2009","journal-title":"Appl. Math. Comput."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1007\/s10589-014-9662-z","article-title":"A family of three-term conjugate gradient methods with sufficient descent property for unconstrained optimization","volume":"60","author":"Narushima","year":"2015","journal-title":"Comput. Optim. Appl."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1007\/BF01593790","article-title":"Restart procedures for the conjugate gradient method","volume":"12","author":"Powell","year":"1977","journal-title":"Math. Program."},{"key":"ref_26","unstructured":"Lootsma, F.A. (1972). A derivative of conjugate gradients. Numerical Methods for Nonlinear Optimization, Academic Press."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1142","DOI":"10.1007\/BF02871976","article-title":"Convergence properties of Beale-Powell restart algorithm","volume":"41","author":"Dai","year":"1998","journal-title":"Sci. China Ser. A Math."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1299","DOI":"10.1080\/10556788.2016.1266354","article-title":"An efficient modified Polak\u2013Ribi\u00e8re\u2013Polyak conjugate gradient method with global convergence properties","volume":"32","author":"Alhawarat","year":"2017","journal-title":"Optim. Methods Softw."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s40096-019-00310-y","article-title":"A globally convergent hybrid conjugate gradient method with strong Wolfe conditions for unconstrained optimization","volume":"14","author":"Kaelo","year":"2019","journal-title":"Math. Sci."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1787","DOI":"10.1080\/00207160.2018.1533122","article-title":"Partially symmetrical derivative-free Liu\u2013Storey projection method for convex constrained equations","volume":"96","author":"Liu","year":"2019","journal-title":"Int. J. Comput. Math."},{"key":"ref_31","first-page":"1","article-title":"An adaptive family of projection methods for constrained monotone non-linear equations with applications","volume":"359","author":"Gao","year":"2019","journal-title":"Appl. Math. Comput."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Zheng, L., Yang, L., and Liang, Y. (2020). A Modified Spectral Gradient Projection Method for Solving Non-linear Monotone Equations with Convex Constraints and Its Application. IEEE Access, 92677\u201392686.","DOI":"10.1109\/ACCESS.2020.2990298"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"112781","DOI":"10.1016\/j.cam.2020.112781","article-title":"A conjugate gradient projection method for solving equations with convex constraints","volume":"375","author":"Zheng","year":"2020","journal-title":"J. Comput. Appl. Math."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"839","DOI":"10.1080\/01630563.2016.1178142","article-title":"Applying the Powell\u2019s Symmetrical Technique to Conjugate Gradient Methods with the Generalized Conjugacy Condition","volume":"37","author":"Benrabia","year":"2016","journal-title":"Numer. Funct. Anal. Optim."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"085008","DOI":"10.1088\/1361-6420\/ab9893","article-title":"Convexification for an inverse parabolic problem","volume":"36","author":"Klibanov","year":"2020","journal-title":"Inverse Probl."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"478","DOI":"10.1137\/070711414","article-title":"A Globally Convergent Numerical Method for a Coefficient Inverse Problem","volume":"31","author":"Beilina","year":"2008","journal-title":"SIAM J. Sci. Comput."},{"key":"ref_37","first-page":"37","article-title":"Non-linear programming, computational methods","volume":"143","author":"Zoutendijk","year":"1970","journal-title":"Integer Nonlinear Program"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1145\/200979.201043","article-title":"CUTE: Constrained and unconstrained testing environment, ACM Trans","volume":"21","author":"Bongartz","year":"1995","journal-title":"Math. Softw."},{"key":"ref_39","unstructured":"(2021, May 20). Available online: http:\/\/users.clas.ufl.edu\/hager\/papers\/Software\/."},{"key":"ref_40","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."},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Alhawarat, A., Salleh, Z., and Masmali, I.A. (2021). A Convex Combination between Two Different Search Directions of Conjugate Gradient Method and Application in Image Restoration. Math. Probl. Eng., 2021.","DOI":"10.1155\/2021\/9941757"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"065011","DOI":"10.1088\/1361-6420\/ab8483","article-title":"The inverse problem of reconstructing reaction\u2013diffusion systems","volume":"36","author":"Kaltenbacher","year":"2020","journal-title":"Inverse Probl."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"115007","DOI":"10.1088\/1361-6420\/ab2aab","article-title":"On the identification of a nonlinear term in a reaction\u2013diffusion equation","volume":"35","author":"Kaltenbacher","year":"2019","journal-title":"Inverse Probl."},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Aver\u00f3s, J.C., Llorens, J.P., and Uribe-Kaffure, R. (2020). Numerical Simulation of Non-Linear Models of Reaction\u2014Diffusion for a DGT Sensor. Algorithms, 13.","DOI":"10.3390\/a13040098"},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Lukyanenko, D., Yeleskina, T., Prigorniy, I., Isaev, T., Borzunov, A., and Shishlenin, M. (2021). Inverse Problem of Recovering the Initial Condition for a Nonlinear Equation of the Reaction\u2013Diffusion\u2013Advection Type by Data Given on the Position of a Reaction Front with a Time Delay. Mathematics, 9.","DOI":"10.3390\/math9040342"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"105824","DOI":"10.1016\/j.cnsns.2021.105824","article-title":"Solving coefficient inverse problems for nonlinear singularly perturbed equations of the reaction-diffusion-advection type with data on the position of a reaction front","volume":"99","author":"Lukyanenko","year":"2021","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1088\/0266-5611\/21\/1\/017","article-title":"Global uniqueness and H\u00f6lder stability for recovering a nonlinear source term in a parabolic equation","volume":"21","author":"Egger","year":"2004","journal-title":"Inverse Probl."}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/14\/8\/227\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:36:11Z","timestamp":1760164571000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/14\/8\/227"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,7,28]]},"references-count":47,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2021,8]]}},"alternative-id":["a14080227"],"URL":"https:\/\/doi.org\/10.3390\/a14080227","relation":{},"ISSN":["1999-4893"],"issn-type":[{"value":"1999-4893","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,7,28]]}}}