{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:14:54Z","timestamp":1760148894502,"version":"build-2065373602"},"reference-count":32,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2023,6,18]],"date-time":"2023-06-18T00:00:00Z","timestamp":1687046400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, we propose a new approach for optimizing a large-scale non-convex differentiable function subject to linear equality constraints. The proposed method, RPCGB (random perturbation of the conditional gradient method with bisection algorithm), computes a search direction by the conditional gradient, and an optimal line search is found by a bisection algorithm, which results in a decrease of the cost function. The RPCGB method is designed to guarantee global convergence of the algorithm. An implementation and testing of the method are given, with numerical results of large-scale problems that demonstrate its efficiency.<\/jats:p>","DOI":"10.3390\/axioms12060603","type":"journal-article","created":{"date-parts":[[2023,6,19]],"date-time":"2023-06-19T02:29:19Z","timestamp":1687141759000},"page":"603","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["RPCGB Method for Large-Scale Global Optimization Problems"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1405-5884","authenticated-orcid":false,"given":"Abderrahmane","family":"Ettahiri","sequence":"first","affiliation":[{"name":"Laboratory LABSI, Faculty of Sciences Agadir (FSA), Ibnou Zohr University, B.P. 8106, Agadir 80000, Morocco"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9947-1408","authenticated-orcid":false,"given":"Abdelkrim","family":"El Mouatasim","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Management, Faculty of Polydisciplinary Ouarzazate (FPO), Ibnou Zohr University, B.P. 284, Ouarzazate 45800, Morocco"}]}],"member":"1968","published-online":{"date-parts":[[2023,6,18]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Frausto Solis, J., Purata Aldaz, J.L., Gonz\u00e1lez del Angel, M., Gonz\u00e1lez Barbosa, J., and Castilla Valdez, G. (2022). SAIPO-TAIPO and Genetic Algorithms for Investment Portfolios. Axioms, 42.","DOI":"10.3390\/axioms11020042"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1016\/j.eneco.2018.12.022","article-title":"Pricing in non-convex markets with quadratic deliverability costs","volume":"80","author":"Kuang","year":"2019","journal-title":"Energy Econ."},{"key":"ref_3","first-page":"565","article-title":"Risk management in portfolio applications of non-convex stochastic programming","volume":"258","author":"Pang","year":"2015","journal-title":"Appl. Math. Comput."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"635","DOI":"10.1007\/s00211-017-0916-4","article-title":"Convex non-convex image segmentation","volume":"138","author":"Chan","year":"2018","journal-title":"Numer. Math."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"332","DOI":"10.1016\/j.jvcir.2013.01.010","article-title":"Non-convex hybrid total variation for image denoising","volume":"24","author":"Oh","year":"2013","journal-title":"J. Vis. Commun. Image Represent."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"2535","DOI":"10.1109\/LCOMM.2018.2875716","article-title":"Semi-blind millimeter-wave channel estimation using atomic norm minimization","volume":"22","author":"Chu","year":"2018","journal-title":"IEEE Commun."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Di Martino, F., and Sessa, S. (2022). A Multilevel Fuzzy Transform Method for High Resolution Image Compression. Axioms, 11.","DOI":"10.3390\/axioms11100551"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Wen, S., Liu, G., Chen, Q., Qu, H., Wang, Y., and Zhou, P. (2019, January 20\u201324). Optimization of precoded FTN signaling with MMSE-based turbo equalization. Proceedings of the IEEE International Conference on Communications (ICC), Shanghai, China.","DOI":"10.1109\/ICC.2019.8761120"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"748","DOI":"10.1016\/j.istruc.2021.11.012","article-title":"Improved arithmetic optimization algorithm and its application to discrete structural optimization","volume":"35","author":"Kaveh","year":"2022","journal-title":"Structures"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"320","DOI":"10.1016\/j.swevo.2018.04.008","article-title":"Adaptive population extremal optimization-based PID neural network for multivariable nonlinear control systems","volume":"44","author":"Zeng","year":"2019","journal-title":"Swarm Evolut. Comput."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1565","DOI":"10.1007\/s11760-020-01696-2","article-title":"Fast gradient descent algorithm for image classification with neural networks","volume":"14","year":"2020","journal-title":"Signal Image Video Process."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"66","DOI":"10.1016\/j.ins.2014.03.059","article-title":"A novel Frank-Wolfe algorithm. Analysis and applications to large-scale SVM training","volume":"285","author":"Nanuclef","year":"2014","journal-title":"Inf. Sci."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Zheng, M., Wang, F., Hu, X., Miao, Y., Cao, H., and Tang, M. (2022). A Method for Analyzing the Performance Impact of Imbalanced Binary Data on Machine Learning Models. Axioms, 11.","DOI":"10.3390\/axioms11110607"},{"key":"ref_14","unstructured":"Berrada, L., Zisserman, A., and Kumar, M.P. (2019, January 6\u20139). Deep Frank-Wolfe for neural network optimization. Proceedings of the International Conference on Learning Representations, New Orleans, LA, USA."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"220","DOI":"10.1007\/978-3-031-24041-6_18","article-title":"Deep Neural Network and YUKI Algorithm for Inner Damage Characterization Based on Elastic Boundary Displacement; Capozucca","volume":"317","author":"Amoura","year":"2023","journal-title":"Lect. Notes Civ. Eng."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"101451","DOI":"10.1016\/j.jocs.2021.101451","article-title":"YUKI Algorithm and POD-RBF for Elastostatic and Dynamic Crack Identification","volume":"55","author":"Benaissa","year":"2021","journal-title":"J. Comput. Sci."},{"key":"ref_17","unstructured":"Moxnes, E. (2015). An Introduction to Deterministic and Stochastic Optimization, Analytical methods for Dynamic Modelers, MIT Press."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1007\/BF01100691","article-title":"Global optimization by random perturbation of the gradient method with a fixed parameter","volume":"5","author":"Pogu","year":"1994","journal-title":"J. Glob. Optim."},{"key":"ref_19","unstructured":"Mandt, S., Hoffman, M., and Blei, D. (2016, January 19\u201324). A variational analysis of stochastic gradient algorithms. Proceedings of the 33rd International Conference on Machine Learning, New York, NY, USA."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"527","DOI":"10.1007\/s10208-015-9296-2","article-title":"Random gradient-free minimization of convex functions","volume":"17","author":"Nesterov","year":"2017","journal-title":"Found. Comput. Math."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Lu, S., Zhao, Z., Huang, K., and Hong, M. (2019, January 12\u201317). Perturbed projected gradient descent converges to approximate second-order points for bound constrained nonconvex problems. Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Brighton, UK.","DOI":"10.1109\/ICASSP.2019.8683241"},{"key":"ref_22","first-page":"142","article-title":"Conditional gradient and bisection algorithms for non-convex optimization problem with random perturbation","volume":"22","author":"Ettahiri","year":"2022","journal-title":"Appl. Math. E-Notes"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1002\/nav.3800030109","article-title":"An Algorithm for Quadratic Programming","volume":"3","author":"Frank","year":"1956","journal-title":"Naval Res. Logist. Q."},{"key":"ref_24","first-page":"1","article-title":"Convergence guarantees for a class of non-convex and non-smooth optimization problems","volume":"20","author":"Khamaru","year":"2019","journal-title":"J. Mach. Learn. Res."},{"key":"ref_25","first-page":"801","article-title":"A multidimensional bisection method for minimizing function over simplex","volume":"2","author":"Baushev","year":"2007","journal-title":"Lect. Notes Eng. Comput. Sci."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"317","DOI":"10.2478\/v10006-011-0024-z","article-title":"Random perturbation of projected variable metric method for linear constraints nonconvex nonsmooth optimization","volume":"21","author":"Ellaia","year":"2011","journal-title":"Int. J. Appl. Math. Comput. Sci."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Bouhadi, M., Ellaia, R., and Souza de Cursi, J.E. (2001). Random perturbations of the projected gradient for linearly constrained problems. Nonconvex Optim. Appl., 487\u2013499.","DOI":"10.1007\/978-1-4613-0279-7_31"},{"key":"ref_28","first-page":"5","article-title":"On the Deng-Lin random number generators and related methods","volume":"14","author":"Touzin","year":"2003","journal-title":"Stat. Comput."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"635","DOI":"10.1007\/s10898-004-9972-2","article-title":"A numerical evaluation of several stochastic algorithms on selected continuous global optimization test problems","volume":"31","author":"Ali","year":"2005","journal-title":"J. Glob. Optim."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1021","DOI":"10.1007\/s11075-018-0471-9","article-title":"A new chaos optimization algorithm based on symmetrization and levelling approaches for global optimization","volume":"79","author":"Aslimani","year":"2018","journal-title":"Numer. Algorithms"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"569","DOI":"10.1016\/j.neucom.2015.04.033","article-title":"An intelligent method of swarm neural networks forequalities constrained nonconvex optimization","volume":"167","author":"Che","year":"2015","journal-title":"Neurocomputing"},{"key":"ref_32","first-page":"10909","article-title":"An extension of the Fletcher Reeves method to linear equality constrained optimization problem","volume":"219","author":"Li","year":"2003","journal-title":"Appl. Math. Comput."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/6\/603\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:55:52Z","timestamp":1760126152000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/6\/603"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,6,18]]},"references-count":32,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2023,6]]}},"alternative-id":["axioms12060603"],"URL":"https:\/\/doi.org\/10.3390\/axioms12060603","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2023,6,18]]}}}