## 2 Theoretical Foundations
### A Collaboration-Based Hybrid GWO-SCA Optimizer  
#### 2.2.1 Gwo  
The grey wolf optimization algorithm was proposed by [PERSON] et al., scholars from Griffith University, in Australia [11]. It belongs to the population intelligence optimization algorithm. The algorithm is inspired by the prey hunting activities of gray wolves. The gray wolf population has a strict hierarchy, with the \(\alpha\) wolf as the leader of the pack, leading the pack's hunting direction and the pack's encirclement. The \(\beta\) wolf serves as the second ranking wolf in the pack and has command over the pack after the alpha wolf. \(\delta\) wolf is the lowest ranking wolf in the pack. The other wolves in the pack are collectively referred to as \(\omega\) wolves, and \(\omega\) wolves can only follow the commands of higher ranked wolves. The hierarchy of gray wolves is shown in Figure 2:  
Individual gray wolves will calculate the distance to their prey as they encircle it, constantly adjusting the position of each individual gray wolf closer to the prey. The distance formula is shown in the following equation:  
\[D=\big{|}C\cdot X_{p}(t)-X(t)\big{|} \tag{5}\]  
where \(D\) implicates the distance vector between the grey wolf and prey, \(C\) is coefficient vector, and \(X_{p}(t),X(t)\) are the positions of the prey and the searching individual gray wolf after iteration times, respectively.  
Grey wolves update their next position based on the distance from their prey:  
\[X(t+1)=X_{p}-A\cdot D \tag{6}\]  
\[a(t)=2(1-t/T_{max}) \tag{7}\]  
\[A=2 ar_{1}-a \tag{8}\]  
\[C=2r_{2} \tag{9}\]  
Figure 2: Gray wolf population hierarchy.  
where \(A\) is the control parameter, \(T_{max}\) is the maximum number of iterations, \(a\) controls the search range of the grey wolf population and is a convergence factor, and \(a\subseteq[0,2]\ r_{1},r_{2}\in[0,1]\) is a random vector.  
During the hunting process, the positions of \(\alpha\) wolf, \(\beta\) wolf and \(\delta\) wolf with high adaptability are used as the direction of action to guide the wolves to the position of the optimal solution, and the positions of the three gray wolves with the highest adaptability are constantly saved in the process of updating the positions every time, so that they can continue to guide the direction of the wolves and determine the orientation of the prey, and gradually approach the prey to guide the wolves to the location of the prey to complete the optimal solution of the optimization process. The process is shown in Figure 3.  
The process of updating the location of wolves is as follows:  
\[\left\{\begin{array}{l}D_{\alpha}=\left|C_{i}\cdot X_{\alpha}-X\right|\\ D_{\beta}=\left|C_{j}\cdot X_{\beta}-X\right|\\ D_{\delta}=\left|C_{k}\cdot X_{\delta}-X\right|\end{array}\right. \tag{10}\]  
\[\left\{\begin{array}{l}X_{i}=X_{\alpha}-A_{i}\cdot D_{\alpha}\\ X_{j}=X_{\beta}-A_{j}\cdot D_{\beta}\\ X_{k}=X_{\delta}-A_{k}\cdot D_{\delta}\end{array}\right. \tag{11}\]  
\[X(t+1)=\frac{X_{i}+X_{j}+X_{k}}{3} \tag{12}\]  
where \(D_{\alpha}\), \(D_{\beta}\), \(D_{\delta}\) are the distances between \(\alpha,\beta,\delta\), respectively, and the searching individual \(\omega\). \(X_{i}\), \(X_{j}\), \(X_{k}\) are the next positions of the searching individual \(\omega\) after being guided by \(\alpha,\beta,\delta\), respectively. \(X_{\alpha}\), \(X_{\beta}\), \(X_{\delta}\) are the positions of the individual \(\alpha,\beta,\delta\) from the prey after t iterations, respectively. \(X\) is the current position of the searching individual. \(A_{i}\), \(C_{i}\) are \(\alpha\) wolf's control parameter, \(A_{j}\), \(C_{j}\) are \(\beta\) wolf's control parameter, and \(A_{k}\), \(C_{k}\) are \(\delta\) wolf's control parameter.