{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:12:56Z","timestamp":1760148776403,"version":"build-2065373602"},"reference-count":63,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2023,6,5]],"date-time":"2023-06-05T00:00:00Z","timestamp":1685923200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"Chinese National Natural Science Foundation","doi-asserted-by":"publisher","award":["12262002","2020GXNSFAA159014","2021GXNSFAA196076"],"award-info":[{"award-number":["12262002","2020GXNSFAA159014","2021GXNSFAA196076"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004607","name":"Natural Science Foundation of Guangxi Province","doi-asserted-by":"publisher","award":["12262002","2020GXNSFAA159014","2021GXNSFAA196076"],"award-info":[{"award-number":["12262002","2020GXNSFAA159014","2021GXNSFAA196076"]}],"id":[{"id":"10.13039\/501100004607","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, a three-dimensional subspace method is proposed, in which the search direction is generated by minimizing the approximation model of the objective function in a three-dimensional subspace. The approximation model of the objective function is not unique, and alternatives can be chosen between a symmetric quadratic model and a conic model by specific criteria. Moreover, the idea of a WLY conjugate gradient method is applied to characterize the change of gradient direction between adjacent iteration points. The strategy of initial stepsize and nonmonotone line search are adopted, and the global convergence of the presented algorithm is established under mild assumptions. In numerical experiments, we use a collection of 80 unconstrained optimization test problems to show the competitive performance of the presented method.<\/jats:p>","DOI":"10.3390\/sym15061207","type":"journal-article","created":{"date-parts":[[2023,6,6]],"date-time":"2023-06-06T01:38:26Z","timestamp":1686015506000},"page":"1207","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["A Three-Dimensional Subspace Algorithm Based on the Symmetry of the Approximation Model and WYL Conjugate Gradient Method"],"prefix":"10.3390","volume":"15","author":[{"given":"Guoxin","family":"Wang","sequence":"first","affiliation":[{"name":"School of Mathematics and Information Science, Guangxi University, Nanning 530004, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shengwei","family":"Yao","sequence":"additional","affiliation":[{"name":"Guangxi (ASEAN) Financial Research Center, Guangxi University of Finance and Economics, Nanning 530007, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mingyang","family":"Pei","sequence":"additional","affiliation":[{"name":"School of Mathematics and Information Science, Guangxi University, Nanning 530004, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jieqiong","family":"Xu","sequence":"additional","affiliation":[{"name":"School of Mathematics and Information Science, Guangxi University, Nanning 530004, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,6,5]]},"reference":[{"key":"ref_1","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. 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