A Novel Modified Lévy Flight Distribution Algorithm based on Nelder-Mead Method for Function Optimization

Year 2021, Volume: 12 Issue: 3, 487 - 497, 29.06.2021

Ahmet Dündar , Davut İzci , Serdar Ekinci , Erdal Eker

https://doi.org/10.24012/dumf.955645

Cited By: 2

Abstract

This paper aims to improve one of the recently proposed metaheuristic approaches known as Lévy flight distribution (LFD) algorithm by adopting a well-known simplex search algorithm named Nelder-Mead (NM) method. Three new strategies were utilized to demonstrate the improved capability of the original LFD algorithm. In the first strategy, NM was run twice as much the number of iterations of LFD after the latter completes its task. In the second strategy, NM was applied after each iterations of LFD instead of waiting for the completion of the latter. Lastly, in the third strategy, NM was applied after each iterations of LFD and run for the total number of current iterations of the latter algorithm. Well-known unimodal and multimodal benchmark functions were adopted, and statistical analysis was performed for performance evaluation. Further assessment was carried out through a nonparametric statistical test. The obtained results have shown the proposed versions of LFD algorithm provide significant performance improvement in general. In addition, the efficiency of the third strategy was found to be better for NM modified LFD algorithm which has greater balance between global and local search stages and can be used as an effective tool for function optimization.

Keywords

Lévy flight distribution, Metaheuristics, Nelder-Mead algorithm, Benchmark functions

References

• [1] L. Abualigah, A. Diabat, S. Mirjalili, M. Abd Elaziz, and A. H. Gandomi, “The Arithmetic Optimization Algorithm,” Comput. Methods Appl. Mech. Eng., vol. 376, p. 113609, Apr. 2021, doi: 10.1016/j.cma.2020.113609.
• [2] W. Zhao, L. Wang, and Z. Zhang, “Supply-Demand-Based Optimization: A Novel Economics-Inspired Algorithm for Global Optimization,” IEEE Access, vol. 7, pp. 73182–73206, 2019, doi: 10.1109/ACCESS.2019.2918753.
• [3] D. Izci, S. Ekinci, E. Eker, and M. Kayri, “Improved Manta Ray Foraging Optimization Using Opposition-based Learning for Optimization Problems,” in 2020 International Congress on Human-Computer Interaction, Optimization and Robotic Applications (HORA), Jun. 2020, pp. 1–6, doi: 10.1109/HORA49412.2020.9152925.
• [4] A. F. Nematollahi, A. Rahiminejad, and B. Vahidi, “A novel meta-heuristic optimization method based on golden ratio in nature,” Soft Comput., vol. 24, no. 2, pp. 1117–1151, 2020, doi: 10.1007/s00500-019-03949-w.
• [5] F. A. Hashim, K. Hussain, E. H. Houssein, M. S. Mabrouk, and W. Al-Atabany, “Archimedes optimization algorithm: a new metaheuristic algorithm for solving optimization problems,” Appl. Intell., 2020, doi: 10.1007/s10489-020-01893-z.
• [6] E. Eker, M. Kayri, S. Ekinci, and D. Izci, “A New Fusion of ASO with SA Algorithm and Its Applications to MLP Training and DC Motor Speed Control,” Arab. J. Sci. Eng., Feb. 2021, doi: 10.1007/s13369-020-05228-5.
• [7] W. Zhao, L. Wang, and Z. Zhang, “Artificial ecosystem-based optimization: a novel nature-inspired meta-heuristic algorithm,” Neural Comput. Appl., pp. 1–43, 2019.
• [8] D. H. Wolpert and W. G. Macready, “No free lunch theorems for optimization,” IEEE Trans. Evol. Comput., vol. 1, no. 1, pp. 67–82, 1997, doi: 10.1109/4235.585893.
• [9] D. Izci and S. Ekinci, “Comparative Performance Analysis of Slime Mould Algorithm For Efficient Design of Proportional–Integral–Derivative Controller,” Electrica, vol. 21, no. 1, pp. 151–159, Jan. 2021, doi: 10.5152/electrica.2021.20077.
• [10] E. H. Houssein, M. R. Saad, F. A. Hashim, H. Shaban, and M. Hassaballah, “Lévy flight distribution: A new metaheuristic algorithm for solving engineering optimization problems,” Eng. Appl. Artif. Intell., vol. 94, p. 103731, 2020, doi: 10.1016/j.engappai.2020.103731.
• [11] S. Ekinci, D. Izci, and B. Hekimoğlu, “Optimal FOPID Speed Control of DC Motor via Opposition-Based Hybrid Manta Ray Foraging Optimization and Simulated Annealing Algorithm,” Arab. J. Sci. Eng., vol. 46, no. 2, pp. 1395–1409, Feb. 2021, doi: 10.1007/s13369-020-05050-z.
• [12] J. A. Nelder and R. Mead, “A Simplex Method for Function Minimization,” Comput. J., vol. 7, no. 4, pp. 308–313, Jan. 1965, doi: 10.1093/comjnl/7.4.308.
• [13] A. Rajan and T. Malakar, “Optimal reactive power dispatch using hybrid Nelder-Mead simplex based firefly algorithm,” Int. J. Electr. Power Energy Syst., vol. 66, pp. 9–24, 2015, doi: 10.1016/j.ijepes.2014.10.041.
• [14] T. Mesbahi, F. Khenfri, N. Rizoug, K. Chaaban, P. Bartholomeüs, and P. Le Moigne, “Dynamical modeling of Li-ion batteries for electric vehicle applications based on hybrid Particle Swarm–Nelder–Mead (PSO–NM) optimization algorithm,” Electr. Power Syst. Res., vol. 131, pp. 195–204, 2016, doi: https://doi.org/10.1016/j.epsr.2015.10.018.
• [15] A. R. Yıldız, B. S. Yıldız, S. M. Sait, S. Bureerat, and N. Pholdee, “A new hybrid Harris hawks-Nelder-Mead optimization algorithm for solving design and manufacturing problems,” Mater. Test., vol. 61, no. 8, pp. 735–743, Aug. 2019, doi: 10.3139/120.111378.
• [16] C. Chen and L. Yu, “A hybrid ant lion optimizer with improved Nelder–Mead algorithm for structural damage detection by improving weighted trace lasso regularization,” Adv. Struct. Eng., vol. 23, no. 3, pp. 468–484, Sep. 2020, doi: 10.1177/1369433219872434.
• [17] H. Zhang, A. A. Heidari, M. Wang, L. Zhang, H. Chen, and C. Li, “Orthogonal Nelder-Mead moth flame method for parameters identification of photovoltaic modules,” Energy Convers. Manag., vol. 211, p. 112764, 2020, doi: https://doi.org/10.1016/j.enconman.2020.112764.
• [18] D. Izci, S. Ekinci, S. Orenc, and A. Demiroren, “Improved Artificial Electric Field Algorithm Using Nelder-Mead Simplex Method for Optimization Problems,” in 2020 4th International Symposium on Multidisciplinary Studies and Innovative Technologies (ISMSIT), Oct. 2020, pp. 1–5, doi: 10.1109/ISMSIT50672.2020.9255255.
• [19] J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM J. Optim., vol. 9, no. 1, pp. 112–147, 1998, doi: 10.1137/S1052623496303470.
• [20] W. Zhao, L. Wang, and Z. Zhang, “Atom search optimization and its application to solve a hydrogeologic parameter estimation problem,” Knowledge-Based Syst., vol. 163, pp. 283–304, 2019, doi: 10.1016/j.knosys.2018.08.030.
• [21] M. Jamil and X. S. Yang, “A literature survey of benchmark functions for global optimisation problems,” Int. J. Math. Model. Numer. Optim., vol. 4, no. 2, pp. 150–194, 2013, doi: 10.1504/IJMMNO.2013.055204.
• [22] W. Zhao, Z. Zhang, and L. Wang, “Manta ray foraging optimization: An effective bio-inspired optimizer for engineering applications,” Eng. Appl. Artif. Intell., vol. 87, p. 103300, Jan. 2020, doi: 10.1016/j.engappai.2019.103300.
• [23] N. Zhang, C. Li, R. Li, X. Lai, and Y. Zhang, “A mixed-strategy based gravitational search algorithm for parameter identification of hydraulic turbine governing system,” Knowledge-Based Syst., vol. 109, pp. 218–237, 2016, doi: 10.1016/j.knosys.2016.07.005.