4.1.1. Search Stage

When the number of iterations is less than two-thirds of the maximum iterations, the aquila individual updates the position by Equation (22) or Equation (24). The specific position update equation is determined by judging the size of the random number. The search stage is as follows:

$$X\_1(t+1) = X\_{\text{best}}(t) \times (1 - \frac{t}{T}) + (X\_M(t) - X\_{\text{best}}(t) \* rand) \tag{22}$$

$$X\_M(t) = \frac{1}{N} \sum\_{i=1}^{N} X\_i(t) \tag{23}$$

where *X*1(*t* + 1) denotes the position of the individual after the update, *t* denotes the current iteration number, *Xbest*(*t*) denotes the position of the best individual, *XM*(*t*) denotes the average of all individual positions, *T* denotes the maximum iteration number, *N* is the number of individuals, and *rand* is a random value between 0 and 1.

$$X\_2(t+1) = X\_{best}(t) \times Levy(D) + X\_R(t) + (y-x) \ast rand \tag{24}$$

where *X*2(*t* + 1) denotes the position of the individual after the update, *XR*(*t*) denotes the position of a random individual in the current population, *D* denotes the dimension of the search space, *Levy* denotes the levy flight function, and *y* and *x* are used to represent the spiral process of the aquila. The corresponding mathematical expressions are as follows:

$$Lcvy(D) = 0.01 \times \frac{\mu \times \delta}{|v|^{\frac{1}{\delta}}} \tag{25}$$

$$\delta = \left( \frac{\Gamma(1+\beta) \times \sin(\frac{\pi\beta}{2})}{\Gamma(\frac{1+\beta}{2}) \times \beta \times 2^{(\frac{\beta-1}{2})}} \right) \tag{26}$$

$$\begin{cases} \ x = r \times \sin(\theta) \\\ y = r \times \cos(\theta) \\\ r = r\_1 + \mathcal{U} \times D\_1 \\\ \theta = -\omega \times D\_1 + \frac{3\pi}{2} \end{cases} \tag{27}$$

where *β* takes the value of 1.5, *u* and *v* are two random numbers taking values between 0 and 1, *r*1 denotes the number of search periods and takes values between 1 and 20, *U* and *ω* are fixed values of 0.0565 and 0.005, respectively, and *D*1 is an integer between 1 and *D*.
