Using MATLAB For Steepest Descent Algorithm (SD) with Conjugate Gradient Algorithm (CG) For Minimizing Unconstrained problems
DOI:
https://doi.org/10.24237/04.02.871Keywords:
Optimization, , Nonlinear, Minimization, Steepest Descent Algorithm, Conjugate Gradient Algorithm, Unconstrained ProblemsAbstract
This paper considers the steepest descent algorithm (SD) and the conjugate gradient algorithm (CG) with a new parameter to minimize nonlinear unconstrained problems. We improve a new parameter for the conjugate gradient algorithm (CG). This new makes the conjugate gradient algorithm (CG) more efficient. Algorithms are submitted and run in the MATLAB program for several examples of minimization problems. The nonlinear functions for unconstrained minimization problems that were used in this paper have not been studied in previous publications. The implementation of the MATLAB program shows that the conjugate gradient algorithm (CG) with new It is more effective than the steepest descent algorithm (SD) while the run time for the steepest descent algorithm (SD) is less than that of the conjugate gradient algorithm (CG). The contribution of this paper will be important to solving more complicated unconstrained nonlinear problems for more variables.
Downloads
References
[1] Arora, R. K., "Optimization: algorithms and applications", 1st ed., CRC press, New York, USA, 2015, pp. 303-310.
[2] Djordjevic, S. S., "Unconstrained Optimization Methods:Conjugate Gradient Methods and Trust-Region", Appl. Math., 2019, doi: 10.5772/intechopen.84374.
[3] E. Carrizosa, E. Conde, M. MuñozMárquez, J. Puerto, "Planarpoint objective location problems with nonconvex constraints: a geometrical construction," J. Global Optim., Vol. 6, no. 1, pp. 77–86, 1995, https://doi.org/10.1007/BF01106606.
[4] E. Carrizosa, J. B. G. Frenk, "Dominating sets for convex functions with some applications," J. Optim. Theo. Appl., Vol. 96, no. 2, pp. 281–29, 1998, https://doi.org/10.1023/A:1022614029984.
[5] Das, I. and Dennis, J. E., "Normal-boundary intersection: A new method for generating the Pareto surface in nonlinear multicriteria optimization problems," SIAM j. on optim., Vol. 8, no. 3, pp. 631-657, 1998, https://doi.org/10.1137/S1052623496307510.
[6] H. Eschenauer, J. Koski, A. Osyczka, Multicriteria Design Optimization, 1st ed., Berlin, Germany, 1990, pp. 71-81, https://doi.org/10.1007/978-3-642-48697-5
[7] G. W. Evans, "An overview of techniques for solving multi-objective mathematical programs," Manag. Sci., Vol. 30, no. 11, pp. 1268–1282, 1984, doi: 10.1287/mnsc.30.11.1268.
[8] D. J. White, "Epsilon-dominating solutions in mean-variance portfolio analysis," European J. Oper. Res., Vol. 105, no. 3, pp. 457–466, 1998.
[9] Curtis, F. E. and Guo, W., "Handling nonpositive curvature in a limited memory steepest descent method," IMA J. of Nume. Anal., Vol. 36, no. 2, pp. 717-742, 2016, doi:10.1093/imanum/drv034.
[10] Drummond, L.G. and Svaiter, B. F., "A steepest descent method for vector optimization," J. of comp. and appl. Math., Vol. 175, no. 2, pp. 395-414, 2005, https://doi.org/10.1016/j.cam.2004.06.018.
[11] Shewchuk, J. R., An introduction to the conjugate gradient method without the agonizing pain, 1994
[12] Fletcher, R., "A limited memory steepest descent method," Math. Progr., Vol. 135, no. 1, pp. 413-436, 2012, https://doi.org/10.1007/s10107-011-0479-6
[13] Hamoda, M. A., Rivaie, M., and Mamat, M., "A new nonlinear conjugate gradient method with exact line search for unconstrained optimization," J. of Bas. Scie., Vol. 30, pp. 1-16 , 2017, https://doi.org/10.59743/jbs.v30i.74
[14] Guo, J., and Wan, Z., "A new three-term conjugate gradient algorithm with modified gradient-differences for solving unconstrained optimization problems," AIMS Math., Vol. 8, no. 2, pp. 2473-2488, 2023, doi:10.3934/math.2023128.
[15] Tang, J., and Wan, Z., "Orthogonal dual graph-regularized nonnegative matrix factorization for co-clustering," J. of Sci. Comput., Vol. 87, no. 3, pp. 66, 2021, https://doi.org/10.1007/s10915-021-01489-w
[16] Li, T., and Wan, Z., "New adaptive barzilai–borwein step size and its application in solving large-scale optimization problems," ANZIAM Journal, Vol. 61, no. 1, pp. 76-98, 2019, https://doi.org/10.1017/S1446181118000263 .
[17] Hu, C., Wan, Z. M., Zhu, S., and Wan, Z., "An integrated stochastic model and algorithm for constrained multi-item newsvendor problems by two-stage decision-making approach," Math. and Comp. in Simul., Vol. 193, pp. 280-300, 2022, https://doi.org/10.1016/j.matcom.2021.10.018.
[18] Wan, Z., and Yu, C., "Reutilization of the existent sales network for recycling unwanted smartphones by nonlinear optimization," J. of Clean. Prod., Vol. 350, 131349, 2022, https://doi.org/10.1016/j.jclepro.2022.131349.
[19] Huang, S., & Wan, Z., "A new nonmonotone spectral residual method for nonsmooth nonlinear equations," J. of Comput. and Appl. Maths., Vol. 313, pp. 82-101, 2017, https://doi.org/10.1016/j.cam.2016.09.014.
[20] Deng, S., Wan, Z., and Chen, X., "An improved spectral conjugate gradient algorithm for nonconvex unconstrained optimization problems," J. of Optimization Theory and appl., Vol. 157, no. 3, pp. 820-842, 2013, http://doi.org/10.1007/s10957-012-0239-7.
[21] Hager, W. W., and Zhang, H., "A new conjugate gradient method with guaranteed descent and an efficient line search," SIAM J. on Optimi., Vol. 16, no. 1, pp. 170-192, 2005, http://doi.org/10.1137/030601880.
[22] Andrei, N., "A simple three-term conjugate gradient algorithm for unconstrained optimization", J. of Comp. and Appl. Math., Vol. 241, pp. 19-29, 2013, http://doi.org/10.1016/j.cam.2012.10.002.
[23] Andrei, N., "On three-term conjugate gradient algorithms for unconstrained optimization," Appl. Math. and Comp., Vol. 219, no. 11, pp. 6316-6327, 2013, http://doi.org/10.1016/j.amc.2012.11.097.
[24] Zhang, L., Zhou, W., and Li, D. H., "A descent modified Polak–Ribière–Polyak conjugate gradient method and its global convergence," IMA J. of Num. Analy., Vol. 26, no. 4, pp. 629-640, 2006, http://doi.org/10.1093/imanum/drl016.
[25] Salih, DTM, and Faraj, BM, "Comparison between steepest descent method and conjugate gradient method by using MATLAB," J. of Stud in Scie. and Engin., Vol. 1, no. 1, pp. 20-31, 2021. https://doi.org/10.53898/josse2021113.
[26] Mahdi, F. Z. A. A., "Investigation of three-term CG-algorithm for non-linear optimization," M.Sc. Thesis, University of Tikrit, Tikrit, Iraq, 2008.
[27] Muhammad, L., Waziri, M. Y., Sulaiman, I. M., Moghrabi, I. A., & Sambas, A., " improved preconditioned conjugate gradient method for unconstrained optimization problem with application in Robot arm control," Eng. Repo., Vol. 6, no. 2, 2024, https://doi.org/10.1002/eng2.12968.
[28] Abbas, I. T., Abd, E. M., & Mohsen, M. J. A., "Using gravitational search algorithm for solving nonlinear regression analysis," Irq. J. of Sci., Vol. 66, no. 3, pp. 1217-1231, 2025, https://doi.org/10.24996/ijs.2025.66.3.20.
[29] Abdulsattar, R., and Abass, I. T., "Tabu search algorithm for solving quadratic assignment problem," In AIP Conference Proceedings, 7 May 2024, https://doi.org/10.1063/5.0209862.
[30] Jasaim, S. H., and Abbas, I. T., "Using waiting queue models to study reducing the phenomenon of delay in one of the health institutions in Baghdad," Irq. J. of Sci., 2025, https://doi.org/10.24996/ijs.2025.66.8.27.
[31] David G. Luenberger Yinyu Ye, Llinear and nonlinear programming, 4th ed., Springer, 2016, pp. 1-11, https://doi.org/10.1007/978-3-030-85450-8
Downloads
Published
Issue
Section
License
Copyright (c) 2026 CC BY 4.0
This work is licensed under a Creative Commons Attribution 4.0 International License.
