{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,10]],"date-time":"2026-02-10T00:49:34Z","timestamp":1770684574816,"version":"3.49.0"},"reference-count":21,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2026,2,1]],"date-time":"2026-02-01T00:00:00Z","timestamp":1769904000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100012190","name":"Ministry of Science and Higher Education of the Russian Federation","doi-asserted-by":"crossref","award":["FEFE-2023-0004"],"award-info":[{"award-number":["FEFE-2023-0004"]}],"id":[{"id":"10.13039\/501100012190","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>Eliminating poor scaling of variables of minimized functions is a pressing issue in solving high-dimensional minimization problems where it is impossible to use methods that change the metric of the space with full-scale metric matrices. In this paper, we propose an iterative method for scaling variables using a diagonal metric matrix and apply it to the gradient minimization method and the conjugate gradient method. In conjugate gradient methods, for quadratic functions, the descent directions are orthogonal to the previous gradient differences. In the proposed method, the transformation of diagonal metric matrices is based on the noted property. For the gradient method with a diagonal metric matrix, an estimate for the convergence rate on strongly convex functions with a Lipschitz gradient was obtained. A computational experiment was conducted, and the presented methods were compared with the Hestenes\u2013Stiefel conjugate gradient method. On the given set of test functions, the gradient method with scaling is comparable in convergence rate to the Hestenes\u2013Stiefel conjugate gradient method, while the conjugate gradient method with scaling matrices significantly outperforms the Hestenes\u2013Stiefel conjugate gradient method. The obtained results confirm the acceleration properties of scaling methods in the case of poor scaling of the variables of the function being minimized. This allows us to conclude that the studied methods can be used alongside conjugate gradient methods to solve smooth, high-dimensional optimization problems with a high degree of conditionality.<\/jats:p>","DOI":"10.3390\/a19020106","type":"journal-article","created":{"date-parts":[[2026,2,2]],"date-time":"2026-02-02T09:00:33Z","timestamp":1770022833000},"page":"106","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Algorithm for Scaling Variables in Minimization Methods"],"prefix":"10.3390","volume":"19","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8500-2050","authenticated-orcid":false,"given":"Elena","family":"Tovbis","sequence":"first","affiliation":[{"name":"Institute of Informatics and Telecommunications, Reshetnev Siberian State University of Science and Technology, 31 Krasnoyarskii Rabochii Prospekt, Krasnoyarsk 660037, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8704-1670","authenticated-orcid":false,"given":"Vladimir","family":"Krutikov","sequence":"additional","affiliation":[{"name":"Institute of Informatics and Telecommunications, Reshetnev Siberian State University of Science and Technology, 31 Krasnoyarskii Rabochii Prospekt, Krasnoyarsk 660037, Russia"},{"name":"Department of Applied Mathematics, Kemerovo State University, 6 Krasnaya Street, Kemerovo 650043, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0667-4001","authenticated-orcid":false,"given":"Lev","family":"Kazakovtsev","sequence":"additional","affiliation":[{"name":"Institute of Informatics and Telecommunications, Reshetnev Siberian State University of Science and Technology, 31 Krasnoyarskii Rabochii Prospekt, Krasnoyarsk 660037, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2026,2,1]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Choi, B., and Park, M. (2025). Minimizing total completion time in single-machine scheduling with convex resource consumption and job rejection. Theor. Comput. Sci., 1064.","DOI":"10.2139\/ssrn.5537647"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Kis, T., and Sz\u00f6gi, E. (2024). Scheduling jobs to minimize a convex function of resource usage. Comput. Oper. Res., 169.","DOI":"10.1016\/j.cor.2024.106748"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Mohan, P., Rajasekaran, V.A., Santhanam, P., Raja, K., Jayagopal, P., Kumar, S., Mallik, S., and Qin, H. (2024). TPEMLB: A novel two-phase energy minimized load balancing scheme for WSN data collection with successive convex approximation using mobile sink. Ain Shams Eng. J., 15.","DOI":"10.1016\/j.asej.2024.102849"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"681","DOI":"10.1016\/j.egyr.2022.10.144","article-title":"Primal\u2013dual interior-point algorithm for electricity cost minimization in a prosumer-based smart grid environment: A convex optimization approach","volume":"8","author":"Gbadega","year":"2022","journal-title":"Energy Rep."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"279","DOI":"10.1016\/j.ins.2021.04.031","article-title":"On Lagrangian L2-norm pinball twin bounded support vector machine via unconstrained convex minimization","volume":"571","author":"Prasad","year":"2021","journal-title":"Inf. Sci."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Zhao, W., and Huang, H. (2024). Adaptive stepsize estimation based accelerated gradient descent algorithm for fully complex-valued neural networks. Expert Syst. Appl., 236.","DOI":"10.1016\/j.eswa.2023.121166"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"66589","DOI":"10.1109\/ACCESS.2022.3184788","article-title":"Polarization-aware prediction of mobile radio wave propagation based on complex-valued and quaternion neural networks","volume":"10","author":"Chen","year":"2022","journal-title":"IEEE Access"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Bagheri, S., Konen, W., and Back, T. (2019, January 13\u201317). Solving Optimization Problems with High Conditioning by Means of Online Whitening. Proceedings of the Genetic and Evolutionary Computation Conference Companion, Prague, Czech Republic.","DOI":"10.1145\/3319619.3322008"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Vaziri, A., and Fang, H. (2025, January 8\u201310). Optimal inferential control of convolutional neural networks. Proceedings of the 2025 American Control Conference (ACC), Denver, CO, USA.","DOI":"10.23919\/ACC63710.2025.11107659"},{"key":"ref_10","unstructured":"Polyak, B.T. (1987). Introduction to Optimization, Optimization Software."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Krutikov, V., Tovbis, E., Stanimirovi\u0107, P., and Kazakovtsev, L. (2023). On the Convergence Rate of Quasi-Newton Methods on Strongly Convex Functions with Lipschitz Gradient. Mathematics, 11.","DOI":"10.3390\/math11234715"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Tovbis, E., Krutikov, V., and Kazakovtsev, L. (2024). Newtonian Property of Subgradient Method with Optimization of Metric Matrix Parameter Correction. Mathematics, 12.","DOI":"10.3390\/math12111618"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Indrapriyadarsini, S., Mahboubi, S., Ninomiya, H., Kamio, T., and Asai, H. (2022). Accelerating Symmetric Rank-1 Quasi-Newton Method with Nesterov\u2019s Gradient for Training Neural Networks. Algorithms, 15.","DOI":"10.20944\/preprints202112.0097.v1"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"503","DOI":"10.1007\/BF01589116","article-title":"On the limited memory BFGS method for large scale optimization","volume":"45","author":"Liu","year":"1989","journal-title":"Math. Program."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1668","DOI":"10.1080\/10556788.2021.1977806","article-title":"Quasi-Newton Methods for Machine Learning: Forget the Past, Just Sample","volume":"37","author":"Berahas","year":"2022","journal-title":"Optim. Methods Softw."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1008","DOI":"10.1137\/140954362","article-title":"A stochastic quasi-Newton method for large-scale optimization","volume":"26","author":"Byrd","year":"2016","journal-title":"SIAM J. Optim."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"785","DOI":"10.1137\/20M1320651","article-title":"Greedy quasi-Newton methods with explicit superlinear convergence","volume":"31","author":"Rodomanov","year":"2021","journal-title":"SIAM J. Optim."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Krutikov, V., Tovbis, E., Gutova, S., Rozhnov, I., and Kazakovtsev, L. (2025). Properties and Application of Incomplete Orthogonalization in the Directions of Gradient Difference in Optimization Methods. Mathematics, 13.","DOI":"10.3390\/math13244036"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"409","DOI":"10.6028\/jres.049.044","article-title":"Method of Conjugate Gradients for Solving Linear Systems","volume":"49","author":"Hestenes","year":"1952","journal-title":"J. Res. Natl. Bur. Stand."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"319","DOI":"10.1016\/j.cnsns.2017.11.013","article-title":"On strong homogeneity of a class of global optimization algorithms working with infinite and infinitesimal scales","volume":"59","author":"Sergeyev","year":"2018","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Shor, N. (1985). Minimization Methods for Nondifferentiable Functions, Springer.","DOI":"10.1007\/978-3-642-82118-9"}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/19\/2\/106\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,2,9]],"date-time":"2026-02-09T08:27:43Z","timestamp":1770625663000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/19\/2\/106"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,2,1]]},"references-count":21,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2026,2]]}},"alternative-id":["a19020106"],"URL":"https:\/\/doi.org\/10.3390\/a19020106","relation":{},"ISSN":["1999-4893"],"issn-type":[{"value":"1999-4893","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,2,1]]}}}