Journal of Global Humanities and Social Sciences 2737-5374 2737-5382 BONI FUTURE DIGITAL PUBLISHING CO.,LIMITED https://ojs.bonfuturepress.com/index.php/GHSS/article/view/1685 5 10 2024 2024-10-30 A Hybrid Conjugate Gradient Algorithm for Nonlinear System of Equations through Conjugacy Condition Aliyu Yusuf,Abdullahi Adamu Kiri,Lukman Lawal,Aliyu Ibrahim Kiri For the purpose of solving a large-scale system of nonlinear equations, a hybrid conjugate gradient algorithm is introduced in this paper, based on the convex combination of βFR k and βPRP k parameters. It is made possible by incorporating the conjugacy condition together with the proposed conjugate gradient search direction. Furthermore, a significant property of the method is that through a non-monotone type line search it gives a descent search direction. Under appropriate conditions, the algorithm establishes its global convergence. Finally, results from numerical tests on a set of benchmark test problems indicate that the method is more effective and robust compared to some existing methods. conjugate gradient parameters, convex combination, conjugacy condition, global convergence, numerical experiments 10.61360/BoniGHSS242016851001

[1] Al-Baali, M. (1985). Descent property and global convergence of the Fletcher-Reeves method with inexact line search. IMA Journal of Numerical Analysis, 5, 121–124. [2] Andrei, N. (2007). Scaled memory-less BFGS preconditioned conjugate gradient algorithm for unconstrained optimization. Optimization Method and Software, 22, 261–571. [3] Andrei, N. (2008). Another hybrid conjugate gradient algorithm for unconstrained optimization. Numerical Algorithm, 47, 143–156. [4] Andrei, N. (2009). Hybrid conjugate gradient algorithm for unconstrained optimization. Journal of Optimization Theory Application, 141, 249–264. [5] Babaie-Kafaki, S., Masoud, F., & Nezam, M. (2011). Two effective hybrid conjugate gradient algorithms based on modified BFGS updates. Numerical Algorithm, 58, 315–331. [6] Dai, Y. H., & Yuan, Y. X. (1999). A nonlinear conjugate gradient with strong global convergence properties. SIAM Journal of Optimization, 10, 177–182. [7] Dai, Y. H., & Yuan, X. Y. (1999). A nonlinear conjugate gradient method with strong global convergence property. SIAM Journal of Optimization, 10, 177–182. [8] Dai, Y. H., & Yuan, Y. X. (1999). A nonlinear conjugate gradient with strong global convergence properties. SIAM Journal on Optimization, 10, 177–182. [9] Dauda, M. K., Mamat, M., Mohamed, M. A., & Waziri, M. Y. (2019). Improved quasi-Newton method via SR1 update for solving symmetric systems of nonlinear equations. Malaysian Journal of Mathematical Sciences, 15(1), 117–130. [10] Dauda, M. K., Mustafa, M., Fatima, S. M., Abubakar, S. M., & Waziri, M. Y. (2019). Derivative-free conjugate gradient method via Broyden’s update for solving symmetric systems of nonlinear equations. Journal of Physics: Conference Series, 1, 1366–1344. [11] Dauda, M. K., Mustafa, M., Waziri, M. Y., Fadhila, A., & Fatima, S. M. (2016). Inexact CG-method via SR1 update for solving systems of nonlinear equations. Far East Journal of Mathematical Science (FJMS), 100(11), 1787–1804. [12] Djordjevic, S. S. (2016). New hybrid conjugate gradient method as a convex combination of FR and PRP methods. Filomat, 30(11), 083–3100. [13] Dolan, E., & Moré, J. (2002). Benchmark optimization software with performance profiles. Journal of Mathematical Program, 91, 201–213. [14] Fang, X., & Ni, Q. (2017). A new derivative-free conjugate gradient method for large-scale nonlinear systems of equations. Bulletin of the Australian Mathematical Society, 95, 500–511. [15] Fletcher, R., & Reeves, C. M. (1964). Function minimization by conjugate gradients. The Computer Journal, 7, 149–154. [16] Fletcher, R., & Reeves, C. M. (1964). Fundamental minimization by conjugate gradients. The Computer Journal, 7, 149–154. [17] Fukushima, M., & Li, D. (1999). A global and super-linear convergent Gauss-Newton base BFGS method for symmetric nonlinear equation. SIAM Journal on Numerical Analysis, 37, 152–172. [18] Hager, W. W., & Zhang, H. (2006). A survey of nonlinear conjugate gradient methods. Pacific Journal of Optimization, 2, 35–58. [19] Halilu, A. S., & Waziri, M. Y. (2017). A transformed double steplength method for solving large-scale systems of nonlinear equations. Journal of Numerical Mathematics and Stochastics, 9(1), 20–32. [20] Hesetenes, M. R., & Stiefel, E. (1952). Methods for conjugate gradients for solving linear systems. Journal of Research of the National Bureau Standards, 49, 409–436. [21] Ioannis, E. L., Vassilis, T., & Panagiotis, P. (2018). A descent hybrid conjugate gradient method based on memoryless BFGS update. Numerical Algorithms, 67, 31–48. [22] Jamilu, S., Waziri, M. Y., & Abba, I. (2017). A new hybrid Dai-Yuan and Hestenes-Stiefel conjugate gradient parameters for solving systems of nonlinear equations. MAYFEB Journal of Mathematics, 1, 44–55. [23] Liu, D. C., & Nocedal, J. (1989). On the limited memory BFGS method for large scale optimization methods. Journal of Mathematical Program, 45, 503–528. [24] Liu, Y., & Storey, C. (1991). Efficient generalized conjugate gradient algorithms, part 1. Journal of Optimization Theory Application, 69, 129–137. [25] Oren, S. S., & Luenberger, D. G. (1974). Self-scaling variable matrix (SSVM) algorithms, part I: Criteria and sufficient conditions for scaling a class of algorithms. Computational Management Science, 20, 845–862. [26] Perry, A. (1978). A modified conjugate gradient algorithm. Operation Research Technology Notes, 26(6), 1073–1078. [27] Polak, E., & Ribiere, G. (1969). Note sur la convergence de methods de direction conjuguees. Revue Francais d’Informatique et de Recherche Operationnelle, 16, 35–43. [28] Powell, M. J. D. (1984). Non-convex minimization calculations and the conjugate gradient method. Numerical Analysis, 1066, 122–141. [29] Waziri, M. Y., Leong, W. J., & Hassan, M. A. (2011). Jacobian free-diagonal Newton’s method for nonlinear systems with singular Jacobian. Malaysian Journal of Mathematical Sciences, 5(2), 241–255. [30] Waziri, M. Y., Leong, W. J., Hassan, M. A., & Monsi, M. (2010). Jacobian computation-free Newton method for systems of nonlinear equations. Journal of Numerical Mathematics and Stochastics, 2, 54–63. [31] Waziri, M. Y., Leong, W. J., Hassan, M. A., & Monsi, M. (2010). A new Newton’s method with diagonal Jacobian approximation for systems of nonlinear equations. Journal of Mathematics and Statistics, 6(3), 246–252. [32] Waziri, M. Y., & Sabiu, J. (2015). A derivative-free conjugate gradient method and its global convergence for symmetric nonlinear equations. Journal of Mathematics and Mathematical Sciences, 8, 2015. [33] Waziri, M. Y., Yusuf, A., & Abubakar, A. B. (2020). Improved conjugate gradient method for nonlinear systems of equations. Computational and Applied Mathematics, 39, 1–17. [34] Zang, L., Zhou, W., & Li, D. (2006). Global convergence of modified Fletcher-Reeves conjugate gradient method with Armijo-type line search. Journal of Numerical Mathematics, 104, 561–572.