Abstract
By drawing an analogy between the population of an evolutionary algorithm and a gas system (which we call a particle system), we first build wave models of evolutionary algorithms based on aerodynamics theory. Then, we solve the models’ linear and quasi-linear hyperbolic equations analytically, yielding wave solutions. These describe the propagation of the particle density wave, which is composed of leftward and rightward waves. We demonstrate the convergence of evolutionary algorithms by analyzing the mechanism underlying the leftward wave, and investigate population diversity by analyzing the rightward wave. To confirm these theoretical results, we conduct experiments that apply three typical evolutionary algorithms to common benchmark problems, showing that the experimental and theoretical results agree. These theoretical and experimental analyses also provide several new clues and ideas that may assist in the design and improvement of evolutionary algorithms.
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References
Back T. Evolutionary Algorithms in Theory and Practice: Evolution Strategies, Evolutionary Programming, Genetic Algorithms. Oxford: Oxford University Press, 1996
Wang F, Zhang H, Li K, et al. A hybrid particle swarm optimization algorithm using adaptive learning strategy. Inf Sci, 2018, 436–437: 162–177
Guo S M, Yang C C. Enhancing differential evolution utilizing eigenvector-based crossover operator. IEEE Trans Evol Computat, 2015, 19: 31–49
Wegener I. Methods for the analysis of evolutionary algorithms on pseudo-boolean functions. In: Evolutionary optimization. Boston: Springer, 2003. 349–369
Beyer H G. Convergence analysis of evolutionary algorithms that are based on the paradigm of information geometry. Evolary Computat, 2014, 22: 679–709
Derrac J, García S, Hui S, et al. Analyzing convergence performance of evolutionary algorithms: a statistical approach. Inf Sci, 2014, 289: 41–58
Tan C J, Neoh S C, Lim C P, et al. Application of an evolutionary algorithm-based ensemble model to job-shop scheduling. J Intell Manuf, 2019, 30: 879–890
Wu H, Kuang L, Wang F, et al. A multiobjective box-covering algorithm for fractal modularity on complex networks. Appl Soft Comput, 2017, 61: 294–313
Goldberg D E, Segrest P. Finite Markov chain analysis of genetic algorithms. In: Proceedings of the 2nd International Conference on Genetic Algorithms, Cambridge, 1987. 1: 1
Rudolph G. Finite Markov chain results in evolutionary computation: a tour d’horizon. Fund Inform, 1998, 35: 67–89
He J, Yao X. Drift analysis and average time complexity of evolutionary algorithms. Artif Intell, 2001, 127: 57–85
Sudholt D. A new method for lower bounds on the running time of evolutionary algorithms. IEEE Trans Evol Computat, 2013, 17: 418–435
Yu Y, Qian C, Zhou Z H. Switch analysis for running time analysis of evolutionary algorithms. IEEE Trans Evol Computat, 2015, 19: 777–792
Bian C, Qian C, Tang K. A general approach to running time analysis of multi-objective evolutionary algorithms. In: Proceedings of 27th International Joint Conference on Artificial Intelligence (IJCAI), Stockholm, 2018. 1405–1411
Mori N, Yoshida J, Tamaki H, et al. A thermodynamical selection rule for the genetic algorithm. In: Proceedings of IEEE International Conference on Evolutionary Computation, Perth, 1995. 1: 188
Cornforth T W, Lipson H. A hybrid evolutionary algorithm for the symbolic modeling of multiple-time-scale dynamical systems. Evol Intel, 2015, 8: 149–164
Li Y X, Zou X F, Kang L S, et al. A new dynamical evolutionary algorithm based on statistical mechanics. J Comput Sci Technol, 2003, 18: 361–368
Li Y X, Xiang Z L, Xia J N. Dynamical system models and convergence analysis for simulated annealing algorithm (in Chinese). Chin J Comput, 2019, 42: 1161–1173
Li Y X, Xiang Z L, Zhang W Y. A relaxation model and time complexity analysis for simulated annealing algorithm (in Chinese). Chin J Comput, 2019. http://kns.cnki.net/kcms/detail/11.1826.TP.20190425.1042.002.html
Zhou Y L. One-Dimensional Unsteady Hydrodynamics. Beijing: Science China Press, 1998
Lamb H. Hydrodynamics. Cambridge: Cambridge University Press, 1993
Gu C H, Li D Q. Mathematical Physics Equations. Beijing: People’s Education Press, 1982
Zhang Y. Expansion Waves and Shock Waves. Beijing: Peking University Press, 1983
Shi Y, Eberhart R C. Empirical study of particle swarm optimization. In: Proceedings of IEEE International Conference on Evolutionary Computation, Washington, 1999. 3: 1945–1950
Liang J J, Qu B Y, Suganthan P N. Problem Definitions and Evaluation Criteria for the CEC 2014 Special Session and Competition on Single Objective Real-Parameter Numerical Optimization. Zhengzhou University and Nanyang Technological University, Technical Report. 2013
Črepinšek M, Liu S H, Mernik M. Exploration and exploitation in evolutionary algorithms. ACM Comput Surv, 2013, 45: 1–33
Liu S H, Mernik M, Bryant B R. To explore or to exploit: an entropy-driven approach for evolutionary algorithms. Int J Knowledge-based Intell Eng Syst, 2009, 13: 185–206
Tang K, Yang P, Yao X. Negatively correlated search. IEEE J Sel Areas Commun, 2016, 34: 542–550
Ursem R K. Diversity-guided evolutionary algorithms. In: Proceedings of International Conference on Parallel Problem Solving from Nature, Berlin, 2002. 462–471
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This work was supported by National Natural Science Foundation of China (Grant No. 61672391).
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Li, Y., Xiang, Z. & Ji, D. Wave models and dynamical analysis of evolutionary algorithms. Sci. China Inf. Sci. 62, 202101 (2019). https://doi.org/10.1007/s11432-018-9852-8
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DOI: https://doi.org/10.1007/s11432-018-9852-8