# HYBRIDIZATION GRADIENT BASED METHODS WITH GENETIC ALGORITHM FOR SOLVING SYSTEMS OF LINEAR EQUATIONS

• AYAD RAMADHAN ALI Dept. of Mathematics, Faculty of Science, University of Zakho, Kurdistan Region-Iraq
• BAYDA GHANIM FATHI Dept. of Mathematics, Faculty of Science, University of Zakho, Kurdistan Region-Iraq
Keywords: Genetic Algorithm, Hybrid Genetic Algorithm, Steepest Descent Method, Cauchy-Barzilai-Borwein Method, Systems of Linear Equations

### Abstract

In this paper, we propose two hybrid gradient based methods and genetic algorithm for solving systems of linear equations with fast convergence. The first proposed hybrid method is obtained by using the steepest descent method and the second one by the Cauchy-Barzilai-Borwein method. These algorithms are based on minimizing the residual of solution which has genetic characteristics. They are compared with the normal genetic algorithm and standard gradient based methods in order to show the accuracy and the convergence speed of them. Since the conjugate gradient method is recommended for solving large sparse and symmetric positive definite matrices, we also compare the numerical results of our proposed algorithms with this method. The numerical results demonstrate the robustness and efficiency of the proposed algorithms. Moreover, we observe that our hybridization of the CBB method and genetic algorithm gives more accurate results with faster convergence than other mentioned methods in all given cases

### References

Bashir, L. Z. (2015). Solve simple linear equation using evolutionary algorithm. World Scientific News, No. 19, 148–167.[1]
Cauchy, A. (1847). Méthode générale pour la résolution des systèms d’équations simultanées. Comput. Rend. Sci. Paris, vol. 25 , 536–538.[2]
Chelouah, R., & Siarry, P. (2003). Genetic and Nelder–Mead algorithms hybridized for a more accurate global optimization of continuous multiminima functions. European Journal of Operational Research, vol. 148 ,No. 2, 335–348. https://doi.org/10.1016/S0377-2217(02)00401-0[3]
Datta, S. (2012). Efficient genetic algorithm on linear programming problem for fittest chromosomes. Journal of Global Research in Computer Science, vol. 3 ,No. 6, 1–7.[4]
El-Emary, I. M. M., & Abd El-Kareem, M. M. (2008). Towards using genetic algorithm for solving nonlinear equation systems. World Applied Sciences Journal, vol. 5 ,No. 3, 282–289.[5]
Fathi, G. B. (2013). Gradient Optimization Algorithms with Fast Convergence. Cardiff University, UK.[6]
Holland, J. H. (1975). Adaptation in natural and artificial systems, univ. of mich. press. Ann Arbor.[7]
Hossain, M., Tanim, A., Choudhury, S., Hayat, S., Kabir, M. N., & Islam, M. M. (2019). An Efficient Solution to Travelling Salesman Problem using Genetic Algorithm with Modified Crossover Operator. EMITTER International Journal of Engineering Technology, vol. 7 ,.[8]
Ikotun, A. M., Akinwale, A. T., & Arogundade, O. T. (2016). Parameter Variation For Linear Equation Solver Using Genetic Algorithm. Journal of Natural Sciences Engineering and Technology, vol. 15 ,No. 2, 42–50.[9]
Ikotun Abiodun, M., Lawal Olawale, N., & Adelokun Adebowale, P. (2011). The effectiveness of genetic algorithm in solving simultaneous equations. International Journal of Computer Applications, vol. 975 , 8887.[10]
Karakatič, S., & Podgorelec, V. (2015). A Survey of Genetic Algorithms for Solving Multi Depot Vehicle Routing Problem. Appl. Soft Comput., vol. 27 ,No. C, 519–532.[11]
Mhetre, P. S. (2012). Genetic algorithm for linear and nonlinear equation. International Journal of Advanced Engineering Technology, vol. 3 ,No. 2, 114–118.[12]
Okamoto, M., Nonaka, T., Ochiai, S., & Tominaga, D. (1998). Nonlinear numerical optimization with use of a hybrid genetic algorithm incorporating the modified Powell method. Applied Mathematics and Computation, vol. 91 ,No. 1, 63–72.[13]
Pal, D., & Parashar, A. (2014). Improved genetic algorithm for intrusion detection system. In 2014 International Conference on Computational Intelligence and Communication Networks (pp. 835–839). IEEE.[14]
Raydan, M., & Svaiter, B. F. (2002). Relaxed steepest descent and Cauchy-Barzilai-Borwein method. Computational Optimization and Applications, vol. 21 ,No. 2, 155–167.[15]
Razali, N. M., & Geraghty, J. (2011). Genetic algorithm performance with different selection strategies in solving TSP. In Proceedings of the world congress on engineering (Vol. 2, pp. 1–6). International Association of Engineers Hong Kong, China.[16]
Renner, G. (2004). Genetic algorithms in computer-aided design. Computer-Aided Design and Applications, vol. 1 ,No. 1–4, 691–700.[17]
Saragih, R. I. E., & Nababan, D. (2019). Increase Performance Genetic Algorithm In Matching System By Setting GA Parameter. In Journal of Physics: Conference Series (Vol. 1175, p. 12100). IOP Publishing.[18]
Sun, L., & Zhang, W. (2006). An accelerated micro genetic algorithm for numerical optimization. In Asia-Pacific Conference on Simulated Evolution and Learning (pp. 277–283). Springer.[19]
Tahk, M., Woo, H., & Park, M. (2007). A Hybrid Optimization Algorithm of Evolutionary Algorithm and Gradient Search. Engineering Optimization, vol. 39 ,No. 1.[20]
Published
2022-11-09
How to Cite
ALI , A. R., & FATHI , B. G. (2022). HYBRIDIZATION GRADIENT BASED METHODS WITH GENETIC ALGORITHM FOR SOLVING SYSTEMS OF LINEAR EQUATIONS . Journal of Duhok University, 25(2), 41-49. https://doi.org/10.26682/sjuod.2022.25.2.4
Section
Pure and Engineering Sciences