Improved grey wolf optimization based on the two-stage search of hybrid CMA-ES

  • Yun-tao ZhaoEmail author
  • Wei-gang Li
  • Ao Liu
Methodologies and Application


Hybrid algorithms with different features are an important trend in algorithm improvement. In this paper, an improved grey wolf optimization based on the two-stage search of hybrid covariance matrix adaptation-evolution strategy (CMA-ES) is proposed to overcome the shortcomings of the original grey wolf optimization that easily falls into the local minima when solving complex optimization problems. First, the improved algorithm divides the whole search process into two stages. In the first stage, the improved algorithm makes full use of the global search ability of grey wolf optimization on a large scale and thoroughly explores the location of the optimal solution. In the second stage, due to CMA-ES having a strong local search capability, the three CMA-ES instances use the α wolf, β wolf and δ wolf as the starting points. In addition, these instances have different step size for parallel local exploitations. Second, in order to make full use of the global search ability of the grey wolf algorithm, the Beta distribution is used to generate as much of an initial population as possible in the non-edge region of the solution space. Third, the new algorithm improves the hunting formula of the grey wolf algorithm, which increases the diversity of the population through the interference of other individuals and reduces the use of the head wolf’s guidance to the population. Finally, the new algorithm is quantitatively evaluated by fifteen standard benchmark functions, five test functions of CEC 2014 suite and two engineering design cases. The results show that the improved algorithm significantly improves the convergence, robustness and efficiency for solving complex optimization problems compared with other six well-known optimization algorithms.


Grey wolf optimization CMA-ES Function optimization Hybrid algorithm Two-stage search 



The authors would like to thank the anonymous reviewers for their valuable comments and suggestions to improve this paper. This work is supported by the National Natural Science Foundation of China [Grant Number 51774219] and the Science and Technology Research Program of Hubei Ministry of Education [Grant Number MADT201706].

Compliance with ethical standards

Conflict of interest

The authors declare that there is no conflict of interest regarding the publication of this paper.

Ethical approval

This article does not contain any studies with animals performed by any of the authors.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Information Science and EngineeringWuhan University of Science and TechnologyWuhanChina
  2. 2.Engineering Research Center for Metallurgical Automation and Measurement Technology of Ministry of EducationWuhanChina
  3. 3.School of ManagementWuhan University of Science and TechnologyWuhanChina

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