International Journal of Nonlinear Analysis and Applications

A new way to obtain fixed point functions using the grey wolf optimizer algorithm

Document Type : Research Paper

Authors

Department of Applied Mathematics, Iran University of Science and Technology, Tehran, Iran

Abstract

In this paper, we introduce a new iterative method for finding the fixed point of a nonlinear function. In fact, we want to offer a new way to obtain the fixed point of various functions using the Grey Wolf Optimizer algorithm. This method is new and very efficient for solving a nonlinear equation. We explain this method with three benchmark functions and compare results with other methods, such as ALO, MVO, MFO and SCA.

Keywords

[1] A.Y. Abdelaziz, E.S. Ali and S.M. Abd Elazim, Flower pollination algorithm and loss sensitivity factors for optimal sizing and placement of capacitors in radial distribution systems, J. Electr. Power Energy Syst. 78 (2016), 207–214.
[2] A. Alizadegan, B. Asady and M. Ahmadpour, Two modified versions of artificial bee colony algorithm, Appl. Math. Comput. 225 (2013), 601–609.
[3] A.A. Alderremy, R.A. Attia, J.F. Alzaidi, D. Lu and M. Khater, Analytical and semi-analytical wave solutions for longitudinal wave equation via modified auxiliary equation method and Adomian decomposition method, Therm. Sci. 23 (2019), 1943–1957.
[4] R.L. Burden and J. Douglas, Faires. Numerical analysis, BROOKS/COLE, 1985.
[5] A. Colorni, M, Dorigo and V. Maniezzo, Distributed optimization by ant colonies, Proc. First Eur. Conf. Artific. Life, 142 (1991), 134–142.
[6] E. Cuevas, M. Cienfuegos, D. Zald´ıvar and M.A. Prez-Cisneros, A swarm optimization algorithm inspired in the behavior of the social-spider, Expert Syst. Appl. 40 (2013), no. 18, 6374–6384.
[7] G. David Edward, Genetic Algorithms in Search, Optimization, and Machine Learning, Addison-Wesle, 2002.
[8] K. Deb, Optimization for engineering design: Algorithms and examples, PHI Learning Pvt. Ltd, 2012.
[9] A. Djerioui, A. Houari, M. Machmoum and M. Ghanes, Grey wolf optimizerbBased Predictive torque control for electric buses applications, Energies 13 (2020), no. 19, 5013.
[10] M. Dorigo, M. Birattari and T. Stutzle, Ant colony optimization, IEEE Comput. Intell. Mag. 1 (2006), no. 4, 28–39.
[11] S. Harifi, M. Khalilian, J. Mohammadzadeh and S. Ebrahimnejad, Emperor Penguins Colony: a new metaheuristic algorithm for optimization, Evol. Intell. 12 (2019), no. 2, 211–226.
[12] J.H. Holland, Adaptation in natural and artifficial systems, Ann Arbor, 1975.
[13] D. Karaboga, An idea based on honey bee swarm for numerical optimization, Technical Report-tr06, Erciyes University, Engineering Faculty, Computer Engineering Department, 2005.
[14] D. Karaboga and B. Basturk, A powerful and efficient algorithm for numerical function optimization: Artificial bee colony (ABC) algorithm, J. Glob. Optim. 39 (2007), no. 3, 459–471.
[15] J. Kennedy and R. Eberhart, Particle swarm optimization, Proc. ICNN’95-Int. Conf. Neural Networks, IEEE, Vol. 4, 1995, pp. 1942–1948.
[16] J. Kennedy, R.C. Eberhart and Y. Shi, Swarm Intelligence, Morgan Kaufmanns Academic Press, San Francisco, 2001.
[17] S.R. Kenneth and R.M. Storn, Differential evolution-a simple and efficient heuristic for global optimization over continuous spaces, J. Glob. Optim. 11 (1997), no. 4, 341–359.
[18] M. Khater, R.A. Attia and D. Lu, Modified auxiliary equation method versus three nonlinear fractional biological models in present explicit wave solutions, Math. Comput. Appl. 24 (2018), no. 1, 1.
[19] M.M. Khater, C. Park, D. Lu and R.A. Attia, Analytical, semi-analytical, and numerical solutions for the Cahn-Allen equation, Adv. Differ. Equ. 2020 (2020), no. 1, 1–12.
[20] H. Ma and D. Simon, Blended biogeography-based optimization for constrained optimization, Eng. Appl. Artif. Intell. 24 (2011), no. 3, 517–525.
[21] P. Mansouri, B. Asady and N. Gupta, The bisection-artificial bee colony algorithm to solve fixed point problems, Appl. Soft Comput. 26 (2015), 143–148.
[22] F. Merrikh-Bayat, The runner-root algorithm: a metaheuristic for solving unimodal and multimodal optimization problems inspired by runners and roots of plants in nature, Appl. Soft Comput. 33 (2015), 292–303.
[23] S. Mirjalili, Dragonfly algorithm: A new meta-heuristic optimization technique for solving single-objective, dis[1]crete, and multi-objective problems, Neural. Comput. Appl. 27 (2016), no. 4, 1053–1073.
[24] S. Mirjalili, Moth-flame optimization algorithm: A novel nature-inspired heuristic paradigm, Knowledge-Based Syst. 89 (2015), 228–249.
[25] S. Mirjalili, The ant lion optimizer, Adv. Eng. Softw. 83 (2015), 80–98.
[26] S. Mirjalili, SCA: A Sine Cosine Algorithm for solving optimization problems, A School of Information and Communication Technology, Griffith University, Nathan Campus, Brisbane, QLD, 4111, 2016.
[27] S. Mirjalili and A. Lewis, The whale optimization algorithm, Adv. Eng. Softw. 95 (2016), 51–67.
[28] S. Mirjalili, S.M. Mirjalili and A. Hatamlou, Multi-verse optimizer: A nature-inspired algorithm for global opti[1]mization, Neural. Comput. Appl. 27 (2016), no. 2, 495–513.
[29] A. Ochoa, L. Margain, A. Hernandez, J. Ponce, A. De Luna, A. Hernandez and O. Castillo, Bat Algorithm to improve a financial trust forest, World Cong. Nature Bio, Inspired Comput., IEEE, 2013, pp. 58–62.
[30] S. Olariu and A.Y. Zomaya, Biology-derived algorithms in engineering optimization, Handbook of Bioinspired Algorithms and Applications, Chapman and Hall/CRC, 2005.
[31] T. Rahkar Farshi, Battle royale optimization algorithm, Neural. Comput. Appl. 33 (2021), no. 4, 1139–1157.
[32] H. Rezazadeh, M. Younis, M. Eslami, M. Bilal and U. Younas, New exact travelling wave solutions to the (2+1)-dimensional Chiral nonlinear Schrodinger equation, Math. Model. Nat. Phenom. 16 (2021), 38.8.
[33] S. Saremi, S. Mirjalili and A. Lewis, Grasshopper optimisation algorithm: Theory and application, Adv. Eng. Software 105 (2017), 30–47.
[34] H. Shah-Hosseini, The intelligent water drops algorithm: a nature-inspired swarm-based optimization algorithm, Int. J. Bio-Inspit. Com. 1 (2009), no. 1-2, 71–79.
[35] J. Vahidi, S.M. Zekavatmand, A. Rezazadeh, M. Inc, M.A. Akinlar and Y.M. Chu, New solitary wave solutions to the coupled Maccaris system, Results Phys. 21 (2021), 103801.
[36] J. Vahidi, S.M. Zekavatmand and H. Rezazadeh, An efficient method for solving hyperbolic partial differential equations, Fifth Nat. Conf. New Approach. Educ. Res., Mahmudabad, 2020.
[37] J. Vahidi, S.M. Zekavatmand and S.M.S. Hejazi, A modern procedure to solve fixed point functions using Bisection[1]Social Spider Algorithm, Sixth Nat. Conf. New Approach. Educ. Res., Mahmudabad, 2021.
[38] J. Vahidi, S.M. Zekavatmand and S.M.S. Hejazi, A novel way to obtain fixed point functions using sine cosine algorithm, Sixth Nat. Conf. New Approach. Educ. Res., Mahmudabad, 2021.
[39] A.M. Wazwaz, A sine-cosine method for handlingnonlinear wave equations, Math. Comput. Model. Dyn. Syst. 40 (2004), no. 5-6, 499–508.
[40] A. M. Wazwaz, The tanh method and the sine-cosine method for solving the KP-MEW equation, Int. J. Comput. Math. 82(2) (2005) 235-246.
[41] X.S. Yang, Nature-inspired metaheuristic algorithms, Luniver Press, 2010.
[42] X. S. Yang, Nature-inspired metaheuristic algorithms. Luniver Press, 2010.
[43] S.W. Yao, S.M. Zekavatmand, H. Rezazadeh, J. Vahidi, M.B. Ghaemi and M. Inc, The solitary wave solutions to the Klein-Gordon-Zakharov equations by extended rational methods, AIP Adv. 11 (2021), no. 6, 065218.
[44] A. Yokus, H. Durur and H. Ahmad, Hyperbolic type solutions for the couple Boiti-Leon-Pempinelli system, FU Math Inf. 35 (2020), no. 2, 523–531.
[45] Y. Yonezawa and T. Kikuchi, Ecological algorithm for optimal ordering used by collective honey bee behavior, MHS’96 Proc. Seventh Int. Symp. Micro Machine Human Sci., IEEE, 1996, pp. 249–256.
[46] S.M. Zekavatmand, H. Rezazadeh and M.B. Ghaemi, Exact travelling wave solutions of nonlinear Cahn-Allen equation, 5th Nat. Conf. Modern Approach. Educ. Res., Mahmudabad, 2020.
[47] S.M. Zekavatmand, H. Rezazadeh and M.B. Ghaemi, Exact travelling wave solutions of nonlinear Cahn-Allen equation, 5th Nat. Conf. Modern Approach. Educ. Res., Mahmudabad, 2020.
[48] S.M. Zekavatmand, J. Vahidi and M.B. Ghaemi, Obtain the fixed point of nonlinear equations through the Whale optimization algorithm, Sixth Nat. Conf. New Approach. Educ. Res., Mahmudabad, 2021.
International Journal of Nonlinear Analysis and Applications
Volume 14, Issue 3
March 2023
Pages 1-7
  • Receive Date: 14 June 2022
  • Accept Date: 23 July 2022