## 3 Methodology
### Global Enhancement Whale Optimization Algorithm
#### 3.2.5 Refraction Learning  
RL is an inverse solution solving method that introduces the refraction principle of light into opposition-based learning (OBL). The superior inverse solution is screened out by OBL, and it is used to replace the existing solution for the next iteration. This strategy has a higher probability of reaching the global optimal solution [37]. However, the conventional OBL is not adjustable in the search range and cannot prevent the inverse solution from falling into the local optimal solution. To fix the defect, a scaling factor is introduced to control the search range according to the refraction principle.  
Figure 4 shows the process of RL, where the center point of the search interval \([x_{\min},x_{\max}]\) is denoted as \(O\). The light propagates from \(P(x,y)\) to the origin \(O\) in one medium and enters another medium. Refraction occurs after it enters a new medium, and the refraction point is \(\hat{P}^{*}\). The distance \(|\overrightarrow{PO}|\) from the incident point to the origin \(O\) is \(h\), and the distance \(|\overrightarrow{OB^{\sharp}}|\) between the origin \(O\) and the refraction point is \(h^{*}\). According to Shell's Law, the relationship between the angle of incidence \(\alpha\) and that of refraction \(\beta\) satisfies the shell formula: \(n=\sin\alpha/\sin\beta\), where \(n\) is the refractive index.  
**Definition 1**.: _The projection \(\hat{x}^{*}\) of the refraction point \(\hat{P}^{*}\) on the x-axis is defined as the inverse solution of \(x\) based on the RL._  
According to the shell formula, we find  
\[\begin{split}\sin\alpha=((x_{min}+x_{max})/2-x)/h\\ \sin\beta=(\hat{x}^{*}-(x_{min}+x_{max})/2)/h^{*}\end{split} \tag{13}\]  
Assuming \(k=h/h^{*}\), (13) can be rewritten as  
\[\hat{x}^{*}=(x_{min}+x_{max})/2+(x_{min}+x_{max})/(2 kn)-x/kn \tag{14}\]  
The inverse solution \(\hat{x}^{*}\) can be solved using (14), where \(k\) is the scaling factor, which is used to adjust the positions of the inverse solutions on the \(x\)-axis. \(n\) is a fine-tuning factor. It can be seen that, assuming \(n=1,k=1\), (14) can be simplified as \(\hat{x}^{*}=-x\), which is the expression for OBL. It can be seen that OBL is a special case of RL. The refractive index of light from air to water is about 0.75. Assuming that \(n=0.75\) is fixed, the change in incident angle causes the change in the projection point \(x^{*}\).  
Figure 3: The curves of an Archimedes spiral and a logarithmic spiral.  
The refraction point can be moved from point \(\hat{P}^{*}\) to point \(\hat{P}_{1}^{*}\) by adjusting \(k\), which is closer to the optimal point. If the \(\hat{P}_{1}^{*}\) does not reach the optimal value, the optimal solution can be reached by adjusting the incidence angle and moving the refraction point to the \(\hat{P}_{2}^{*}\). In order to improve the projection range of the inverse solution, we set \(k\) in a linearly decreasing trend:  
\[k=k_{min}+(k_{max}-k_{min})t/t_{max} \tag{15}\]  
The \(N\)-dimensional positions of a whale after position updating are denoted as:  
\[\widetilde{\mathbf{X}}=(\widetilde{x}_{0},\widetilde{x}_{1},\ldots, \widetilde{x}_{i},\ldots,\widetilde{x}_{N-1}) \tag{16}\]  
where \(x_{i,j}\in[x_{min},x_{max}],i=0,1,\ldots,N-1\). \(x_{min}\) and \(x_{max}\) are the minimum and maximum values in the search range, respectively. The inverse solution \(\hat{\mathbf{X}}^{*}\) obtained by RL is  
\[\hat{\mathbf{X}}^{*}=\left(\hat{x}_{0}^{*},\hat{x}_{1}^{*},\ldots,\hat{x}_{i }^{*},\ldots,\hat{x}_{N-1}^{*}\right) \tag{17}\]  
If \(\mathbf{PSLL}(\hat{\mathbf{X}}^{*})<\mathbf{PSLL}(\widetilde{\mathbf{X}})\), \(\hat{\mathbf{X}}^{*}\) is chosen as the best candidate solution. Otherwise, \(\widetilde{\mathbf{X}}\) is chosen as the best candidate solution.