Hybridizing grey wolf optimization with neural network algorithm for global numerical optimization problems


This paper proposes a novel hybrid algorithm, called grey wolf optimization with neural network algorithm (GNNA), for solving global numerical optimization problems. The core idea of GNNA is to make full use of good global search ability of neural network algorithm (NNA) and fast convergence of grey wolf optimizer (GWO). Moreover, both NNA and GWO are improved to boost their own advantages. For NNA, an improved NNA is given to strengthen the exploration ability of NNA by discarding transfer operator and introducing random modification factor. For GWO, an enhanced GWO is presented, which adjusts the exploration rate based on reinforcement learning principles. Then the improved NNA and the enhanced GWO are hybridized by dynamic population mechanism. A comprehensive set of 23 well-known unconstrained benchmark functions are employed to examine the performance of GNNA compared with 13 metaheuristic algorithms. Such comparisons suggest that the combination of the improved NNA and the enhanced GWO is very effective and GNNA is clearly seen to be more successful in both solution quality and computational efficiency.

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  1. 1.

    Savsani P, Savsani V (2016) Passing vehicle search (PVS): A novel metaheuristic algorithm. Appl Math Model 40:3951–3978. https://doi.org/10.1016/j.apm.2015.10.040

    Article  MATH  Google Scholar 

  2. 2.

    Zhang J, Xiao M, Gao L, Pan Q (2018) Queuing search algorithm: a novel metaheuristic algorithm for solving engineering optimization problems. Appl Math Model 63:464–490. https://doi.org/10.1016/j.apm.2018.06.036

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Rao RV, Savsani VJ, Vakharia DP (2012) Teaching–learning-based optimization: an optimization method for continuous non-linear large scale problems. Inf Sci 183:1–15. https://doi.org/10.1016/j.ins.2011.08.006

    MathSciNet  Article  Google Scholar 

  4. 4.

    Eskandar H, Sadollah A, Bahreininejad A, Hamdi M (2012) Water cycle algorithm—a novel metaheuristic optimization method for solving constrained engineering optimization problems. Comput Struct 110–111:151–166. https://doi.org/10.1016/j.compstruc.2012.07.010

    Article  Google Scholar 

  5. 5.

    Kennedy J, Eberhart R (1995) Particle swarm optimization. In: Proceedings of ICNN’95—international conference on neural networks, vol 4, pp 1942–1948

  6. 6.

    Yang X, Deb S (2009) Cuckoo search via Lévy flights. In: 2009 world congress on nature biologically inspired computing (NaBIC), pp 210–214

  7. 7.

    Yang X-S (2010) A new metaheuristic bat-inspired algorithm. In: González JR, Pelta DA, Cruz C et al (eds) Nature inspired cooperative strategies for optimization (NICSO 2010). Springer, Berlin, pp 65–74

    Google Scholar 

  8. 8.

    Mirjalili S, Mirjalili SM, Lewis A (2014) Grey wolf optimizer. Adv Eng Softw 69:46–61. https://doi.org/10.1016/j.advengsoft.2013.12.007

    Article  Google Scholar 

  9. 9.

    Gandomi AH, Yang X-S, Alavi AH (2011) Mixed variable structural optimization using firefly algorithm. Comput Struct 89:2325–2336. https://doi.org/10.1016/j.compstruc.2011.08.002

    Article  Google Scholar 

  10. 10.

    Mirjalili S, Gandomi AH, Mirjalili SZ et al (2017) Salp swarm algorithm: a bio-inspired optimizer for engineering design problems. Adv Eng Softw 114:163–191. https://doi.org/10.1016/j.advengsoft.2017.07.002

    Article  Google Scholar 

  11. 11.

    Mirjalili S, Lewis A (2016) The whale optimization algorithm. Adv Eng Softw 95:51–67. https://doi.org/10.1016/j.advengsoft.2016.01.008

    Article  Google Scholar 

  12. 12.

    Goldberg DE, Holland JH (1988) Genetic algorithms and machine learning. Mach Learn 3:95–99. https://doi.org/10.1023/A:1022602019183

    Article  Google Scholar 

  13. 13.

    Rahnamayan S, Tizhoosh HR, Salama MMA (2008) Opposition-based differential evolution. IEEE Trans Evol Comput 12:64–79. https://doi.org/10.1109/TEVC.2007.894200

    Article  Google Scholar 

  14. 14.

    Simon D (2008) Biogeography-based optimization. IEEE Trans Evol Comput 12:702–713. https://doi.org/10.1109/TEVC.2008.919004

    Article  Google Scholar 

  15. 15.

    Rao RV, Savsani VJ, Vakharia DP (2011) Teaching–learning-based optimization: a novel method for constrained mechanical design optimization problems. Comput Aided Des 43:303–315. https://doi.org/10.1016/j.cad.2010.12.015

    Article  Google Scholar 

  16. 16.

    Mirjalili S (2016) SCA: a sine cosine algorithm for solving optimization problems. Knowl Based Syst 96:120–133. https://doi.org/10.1016/j.knosys.2015.12.022

    Article  Google Scholar 

  17. 17.

    Sadollah A, Sayyaadi H, Yadav A (2018) A dynamic metaheuristic optimization model inspired by biological nervous systems: neural network algorithm. Appl Soft Comput 71:747–782. https://doi.org/10.1016/j.asoc.2018.07.039

    Article  Google Scholar 

  18. 18.

    Kirkpatrick S, Gelatt CD, Vecchi MP (1983) Optimization by simulated annealing. Science 220:671–680. https://doi.org/10.1126/science.220.4598.671

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Wolpert DH, Macready WG (1997) No free lunch theorems for optimization. IEEE Trans Evol Comput 1:67–82. https://doi.org/10.1109/4235.585893

    Article  Google Scholar 

  20. 20.

    Zang H, Zhang S, Hapeshi K (2010) A review of nature-inspired algorithms. J Bionic Eng 7:S232–S237. https://doi.org/10.1016/S1672-6529(09)60240-7

    Article  Google Scholar 

  21. 21.

    Garg H (2019) A hybrid GSA-GA algorithm for constrained optimization problems. Inf Sci 478:499–523. https://doi.org/10.1016/j.ins.2018.11.041

    Article  Google Scholar 

  22. 22.

    Xiong G, Zhang J, Yuan X et al (2018) Parameter extraction of solar photovoltaic models by means of a hybrid differential evolution with whale optimization algorithm. Sol Energy 176:742–761. https://doi.org/10.1016/j.solener.2018.10.050

    Article  Google Scholar 

  23. 23.

    Le DT, Bui D-K, Ngo TD et al (2019) A novel hybrid method combining electromagnetism-like mechanism and firefly algorithms for constrained design optimization of discrete truss structures. Comput Struct 212:20–42. https://doi.org/10.1016/j.compstruc.2018.10.017

    Article  Google Scholar 

  24. 24.

    Qais MH, Hasanien HM, Alghuwainem S (2018) Augmented grey wolf optimizer for grid-connected PMSG-based wind energy conversion systems. Appl Soft Comput 69:504–515. https://doi.org/10.1016/j.asoc.2018.05.006

    Article  Google Scholar 

  25. 25.

    Long W, Liang X, Cai S et al (2017) A modified augmented Lagrangian with improved grey wolf optimization to constrained optimization problems. Neural Comput Appl 28:421–438. https://doi.org/10.1007/s00521-016-2357-x

    Article  Google Scholar 

  26. 26.

    Khairuzzaman AKM, Chaudhury S (2017) Multilevel thresholding using grey wolf optimizer for image segmentation. Expert Syst Appl 86:64–76. https://doi.org/10.1016/j.eswa.2017.04.029

    Article  Google Scholar 

  27. 27.

    Sahoo A, Chandra S (2017) Multi-objective grey wolf optimizer for improved cervix lesion classification. Appl Soft Comput 52:64–80. https://doi.org/10.1016/j.asoc.2016.12.022

    Article  Google Scholar 

  28. 28.

    Zhang X, Kang Q, Cheng J, Wang X (2018) A novel hybrid algorithm based on biogeography-based optimization and grey wolf optimizer. Appl Soft Comput 67:197–214. https://doi.org/10.1016/j.asoc.2018.02.049

    Article  Google Scholar 

  29. 29.

    Lu C, Gao L, Li X, Xiao S (2017) A hybrid multi-objective grey wolf optimizer for dynamic scheduling in a real-world welding industry. Eng Appl Artif Intell 57:61–79

    Article  Google Scholar 

  30. 30.

    Emary E, Zawbaa HM, Grosan C (2018) Experienced gray wolf optimization through reinforcement learning and neural networks. IEEE Trans Neural Netw Learn Syst 29:681–694. https://doi.org/10.1109/TNNLS.2016.2634548

    MathSciNet  Article  Google Scholar 

  31. 31.

    Rakhshani H, Rahati A (2017) Snap-drift cuckoo search: a novel cuckoo search optimization algorithm. Appl Soft Comput 52:771–794. https://doi.org/10.1016/j.asoc.2016.09.048

    Article  MATH  Google Scholar 

  32. 32.

    Kaelbling LP, Littman ML, Moore AP (1996) Reinforcement learning: a survey. J Artif Intell Res 4:237–285

    Article  Google Scholar 

  33. 33.

    Salgotra R, Singh U, Saha S (2018) New cuckoo search algorithms with enhanced exploration and exploitation properties. Expert Syst Appl 95:384–420. https://doi.org/10.1016/j.eswa.2017.11.044

    Article  Google Scholar 

  34. 34.

    Sun Y, Wang X, Chen Y, Liu Z (2018) A modified whale optimization algorithm for large-scale global optimization problems. Expert Syst Appl 114:563–577. https://doi.org/10.1016/j.eswa.2018.08.027

    Article  Google Scholar 

  35. 35.

    Long W, Jiao J, Liang X, Tang M (2018) An exploration-enhanced grey wolf optimizer to solve high-dimensional numerical optimization. Eng Appl Artif Intell 68:63–80. https://doi.org/10.1016/j.engappai.2017.10.024

    Article  Google Scholar 

  36. 36.

    Wang H, Wu Z, Rahnamayan S et al (2011) Enhancing particle swarm optimization using generalized opposition-based learning. Spec Issue Interpret Fuzzy Syst 181:4699–4714. https://doi.org/10.1016/j.ins.2011.03.016

    MathSciNet  Article  Google Scholar 

  37. 37.

    Derrac J, García S, Molina D, Herrera F (2011) A practical tutorial on the use of nonparametric statistical tests as a methodology for comparing evolutionary and swarm intelligence algorithms. Swarm Evol Comput 1:3–18. https://doi.org/10.1016/j.swevo.2011.02.002

    Article  Google Scholar 

  38. 38.

    Mafarja M, Aljarah I, Heidari AA et al (2018) Binary dragonfly optimization for feature selection using time-varying transfer functions. Knowl Based Syst 161:185–204. https://doi.org/10.1016/j.knosys.2018.08.003

    Article  Google Scholar 

  39. 39.

    Sun G, Ma P, Ren J et al (2018) A stability constrained adaptive alpha for gravitational search algorithm. Knowl Based Syst 139:200–213. https://doi.org/10.1016/j.knosys.2017.10.018

    Article  Google Scholar 

  40. 40.

    Martínez-Peñaloza M-G, Mezura-Montes E (2018) Immune generalized differential evolution for dynamic multi-objective environments: an empirical study. Knowl Based Syst 142:192–219. https://doi.org/10.1016/j.knosys.2017.11.037

    Article  Google Scholar 

  41. 41.

    Yi J, Gao L, Li X et al (2019) An on-line variable-fidelity surrogate-assisted harmony search algorithm with multi-level screening strategy for expensive engineering design optimization. Knowl Based Syst 170:1–19. https://doi.org/10.1016/j.knosys.2019.01.004

    Article  Google Scholar 

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Correspondence to Zhigang Jin.

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Zhang, Y., Jin, Z. & Chen, Y. Hybridizing grey wolf optimization with neural network algorithm for global numerical optimization problems. Neural Comput & Applic 32, 10451–10470 (2020). https://doi.org/10.1007/s00521-019-04580-4

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  • Artificial neural networks
  • Reinforcement learning
  • Grey wolf optimizer
  • Numerical optimization