SCIENCE CHINA Information Sciences, Volume 62, Issue 10: 202101(2019) https://doi.org/10.1007/s11432-018-9852-8

## Wave models and dynamical analysis of evolutionary algorithms

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• ReceivedOct 18, 2018
• AcceptedMar 29, 2019
• PublishedSep 3, 2019
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### 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.

### Acknowledgment

This work was supported by National Natural Science Foundation of China (Grant No. 61672391).

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• Figure 1

(Color online) Illustration showing (a) the rightward linear wave and (b) clusters of characteristic lines.

• Figure 2

(Color online) Wave propagation for $f(x)$, showing the (a) particle density distribution, (b) volume elasticity coefficient $\kappa$, and (c) wave propagation speed $a$ over time.

• Figure 3

(Color online) (a) ${\rm~TSP}(n=50)$; (b) ${\rm~TSP}(n=100)$.

• Figure 4

(Color online) PSO applied to (a) $f_1$ and (b) $f_2$.

• Figure 5

(Color online) DE applied to (a) $f_1$ and (b) $f_2$.

• Figure 6

(Color online) PSO vs. DE, applied to (a) $f_1$ and (b) $f_2$.

• Figure 7

(Color online) $a$ vs. $a_{sw}$, for (a) $f_1$ and (b) $f_2$.

• Table 1   Test functions used for PSO and DE
 Test function $D$ Search range $f_{\rm~min}$ Name $f_1(y)=\sum_{i=1}^{D-1}(100(y_{i}^2-y_{i+1}^2)+(y_i-1)^2)+F_{4}^*$, 30 $[-100,100]^{D}$ 400 Shifted and Rotated where $y=M(\frac{2.048z}{100})+1$, $M$ is a rotation matrix, Rosenbrock [25] $z=x-o$, and $o=[o_1,o_2,\ldots,o_n]$ is the shifted global optimum. $f_2(y)=\sum_{i=1}^D\frac{y_{i}^2}{4000}-\prod_{i=1}^D{\rm~cos}(\frac{y_i}{\sqrt~i})+1+F_{7}^*$, 30 $[-100,100]^{D}$ 700 Shifted and Rotated where $y=M(\frac{600z}{100})$, $M$ is a rotation matrix, Griewank [25] $z=x-o$, and $o=[o_1,o_2,\ldots,o_n]$ is the shifted global optimum.

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