## 3 Improved Grey Wolf Particle Swarm Hybrid Optimization MSVR Localization Model Construction
### Grey Wolf Optimization Algorithm  
The Grey Wolf Optimization (GWO) algorithm draws inspiration from the predatory behavior and social hierarchy observed in grey wolves [44]. Figure 3 depicts this societal structure of a wolf pack. At the pinnacle are the \(\alpha\) wolves, who lead the pack in hunting and decision-making. The \(\beta\) wolves form the second tier, supporting and assisting the \(\alpha\) wolves in their endeavors. Next in the hierarchy are the \(\delta\) wolves, responsible for transmitting the decisions of the higher-ranking wolves. Finally, the \(\omega\) wolves, situated at the lowest tier, follow the commands of the other members. This collaborative structure enables the wolf pack to effectively carry out its hunting activities.  
The primary objective of the Grey Wolf Optimization algorithm is to obtain the optimal solution for parameters, analogous to capturing prey. Therefore, before the hunt begins, the wolf pack first determines an approximate location of the prey, followed by encirclement. The mathematical model for this behavior is expressed as:  
\[\mathbf{d}=\left|\mathbf{C}\times\mathbf{X}_{p}(i)-\mathbf{X}_{\delta}(i)\right| \tag{17}\]  
\[\mathbf{X}_{\delta}(i+1)=\mathbf{X}_{p}(i)-\mathbf{A}\times\mathbf{d} \tag{18}\]  
In the equation above, \(\mathbf{X}_{p}(i)\) and \(\mathbf{X}_{\delta}(i)\) denote the positions vectors of the prey and the grey wolf at the \(i\)-th iteration, respectively. \(\mathbf{d}\) is the distance between them, while \(\mathbf{A}\) and \(\mathbf{C}\) are coefficient vectors, which are determined as follows:  
\[\left\{\begin{array}{l}\mathbf{A}=2\mathbf{a}\cdot\mathbf{r}_{1}-\mathbf{a}\\ \mathbf{C}=2\mathbf{r}_{2}\\ \mathbf{a}=2(1-i/I_{\max})\end{array}\right. \tag{19}\]  
where \(\mathbf{r}_{1}\) and \(\mathbf{r}_{2}\) are random vectors within [0, 1], \(\mathbf{I}_{\max}\) represents the maximum number of iterations, and \(\mathbf{a}\) decreases linearly from 2 to 0 as the number of iterations increases. Once the wolves complete their encirclement of the prey, they initiate an attack. Depending on the prey's location, the positions of the grey wolves at all hierarchical levels will adjust. The specific updated strategies are as follows:  
\[\left\{\begin{array}{l}\mathbf{d}_{\mathbf{a}}=\left|\mathbf{C}_{1}\times\mathbf{X}_{\pi}(i)- \mathbf{X}_{p}(i)\right|\\ \mathbf{d}_{\beta}=\left|\mathbf{C}_{1}\times\mathbf{X}_{\beta}(i)-\mathbf{X}_{p}(i)\right|\\ \mathbf{d}_{\delta}=\left|\mathbf{C}_{1}\times\mathbf{X}_{\delta}(i)-\mathbf{X}_{p}(i)\right| \end{array}\right. \tag{20}\]  
\[\left\{\begin{array}{l}\mathbf{X}_{1}(i)=\left|\mathbf{X}_{\alpha}(i)-\mathbf{A}_{1} \times\mathbf{d}_{\alpha}\right|\\ \mathbf{X}_{2}(i)=\left|\mathbf{X}_{\beta}(i)-\mathbf{A}_{2}\times\mathbf{d}_{\beta}\right|\\ \mathbf{X}_{3}(i)=\left|\mathbf{X}_{\delta}(i)-\mathbf{A}_{3}\times\mathbf{d}_{\delta}\right| \end{array}\right. \tag{21}\]  
\[\mathbf{X}(i+1)=\frac{\mathbf{X}_{1}(i)+\mathbf{X}_{2}(i)+\mathbf{X}_{3}(i)}{3} \tag{22}\]  
Using Equation (20), the distances of other wolves from \(\alpha\), \(\beta\), and \(\delta\) are calculated. Equation (21) determines the moving directions of the remaining grey wolf individuals. Their positions are then updated according to Equation (22). As the iterations proceed, the wolf pack eventually completes the hunt, with the final position of the \(\alpha\) wolf signifying the optimal parameters. Figure 4 illustrates the optimization process of the Grey Wolf algorithm.  
Figure 3: Social class in the grey wolf population.