## 3 PSO Based on ABC Search (F-WAPSO)
### Improved PSO and Its Convergence Analysis
#### 3.1.2 Convergence Analysis of WASP  
**Theorem 1.**_Each particle converges to a stable point \(p_{i}\), where \(p_{i}=\frac{\phi_{i}\textit{pbest}+\phi_{2}\textit{best}}{\phi_{1}+\phi_{2}}, \lim\limits_{t\rightarrow+\infty}x_{i}(\tau)=p_{i}\)._  
**Proof of Theorem 1.** The velocity and position update formulas are as follows:  
\[v(\tau+1)=\omega v(\tau)+\phi_{1}(\textit{pbest}-x(\tau))+\phi_{2}(\textit{ gbest}-x(\tau)) \tag{8}\]  
\[x(\tau+1)=x(\tau)+v(\tau+1) \tag{9}\]  
Substituting Equation (8) into Equation (9) gives Equation (10):  
\[x_{\tau+1}=(1+\omega-\phi_{1}-\phi_{2})x_{\tau}-\omega x_{\tau-1}+\phi_{1} \textit{pbest}+\phi_{2}\textit{gbest} \tag{10}\]  
Equation (10) can be converted to:  
\[\begin{bmatrix}x_{\tau+1}\\ x_{\tau}\\ 1\end{bmatrix}=\begin{bmatrix}1+\omega-\phi_{1}-\phi_{2}&-\omega&\phi_{1} \textit{pbest}+\phi_{2}\textit{gbest}\\ 1&0&0\\ 0&0&1\end{bmatrix}\begin{bmatrix}x_{\tau}\\ x_{\tau-1}\\ 1\end{bmatrix} \tag{11}\]  
The characteristic polynomial of Equation (11) is as follows:  
\[(1-\lambda)\Big{(}\lambda^{2}-\lambda(1+\omega-\phi_{1}-\phi_{2})+\omega \Big{)}=0.\]  
The characteristic solution can be written as follows:  
\[\lambda_{1}=\frac{1+\omega-\phi_{1}-\phi_{2}+\varepsilon}{2},\]  
\[\lambda_{2}=\frac{1+\omega-\phi_{1}-\phi_{2}-\varepsilon}{2},\]  
where \(\varepsilon=\sqrt{\left(1+\omega-\phi_{1}-\phi_{2}\right)^{2}-4\omega}\).  
Under initial conditions \(x_{0}=x(0)\) and \(x_{1}=x(1)\), the recurrence relation is:  
\[x_{\tau}=k_{1}+k_{2}\lambda_{1}^{\tau}+k_{3}\lambda_{2}^{\tau} \tag{12}\]  
where \(k_{1}=\frac{\phi_{1}\textit{pbest}+\phi_{2}\textit{best}}{\phi_{1}+\phi_{2}}\), \(k_{2}=\frac{\lambda_{2}(x_{0}-x_{1})-x_{1}+x_{2}}{\varepsilon(\lambda_{1}-1)}\), \(k_{3}=\frac{\lambda_{1}(x_{1}-x_{0})+x_{1}-x_{2}}{\varepsilon(\lambda_{2}-1)}\).  
The convergence of the particle can be expressed as follows:  
\[\lim\limits_{\tau\rightarrow+\infty}x_{\tau}=k_{1}=\frac{\phi_{1}\textit{pbest }+\phi_{2}\textit{gbest}}{\phi_{1}+\phi_{2}} \tag{13}\]  
For any complex number \(z\), \(z^{\tau}\) can be written as follows:  
\[z^{\tau}=\Big{(}\|z\|^{\textit{\theta}}\Big{)}^{\tau}\]  
\[z^{\tau}=\|z\|^{\tau}(\cos(\theta\tau)+l\sin(\theta\tau))\]  
where \(\theta=\arg(z)\). Then, the convergence is expressed as follows:  
\[\lim\limits_{\tau\rightarrow+\infty}z^{\tau}=\lim\limits_{\tau\rightarrow+ \infty}\big{(}\|z\|^{\tau}(\cos(\theta\tau)+l\sin(\theta\tau))\big{)} \tag{14}\]  
When \(\|z\|<1\), then \(\lim\limits_{\tau\rightarrow+\infty}z^{\tau}=0\), \(\|\lambda_{1}\|>1\) or \(\|\lambda_{2}\|>1\), the system diverges, and \(\lim\limits_{\tau\rightarrow+\infty}(k_{1}+k_{2}\lambda_{1}^{\tau}+k_{3}\lambda_ {2}^{\tau})\) does not exist.  
When \(\|z\|=1\), \(\underset{\tau\rightarrow+\infty}{\lim}z^{\tau}=\underset{\tau\rightarrow+ \infty}{\lim}(1^{\tau}(\cos(\theta\tau)+l\sin(\theta\tau)))\) does not exist, and \(\|\lambda_{1}\|=1\) or \(\|\lambda_{2}\|=1\), and the system diverges.  
When \(\|\lambda_{1}\|<1\) and \(\|\lambda_{2}\|<1\), the system diverges. Because \(\|\lambda_{1}\|<1\), \(\underset{\tau\rightarrow+\infty}{\lim}\lambda_{1}^{\tau}=0\) and because \(\|\lambda_{2}\|<1\), \(\underset{\tau\rightarrow+\infty}{\lim}\lambda_{2}^{\tau}=0\); then, \(\underset{\tau\rightarrow+\infty}{\lim}x_{\tau}=k_{1}=\frac{\phi_{1}p_{best}+ \phi_{2}p_{best}}{\phi_{1}+\phi_{2}}\).  
This satisfies the proof. \(\square\)  
Conclusion: When \(\|\lambda_{1}\|<1\) and \(\|\lambda_{2}\|<1\), the particle or particles swarm converges to a weighted average of its individual and global optimal positions. The convergence point is \(p_{i}=\frac{\phi_{1}p_{best}+\phi_{2}p_{best}}{\phi_{1}+\phi_{2}}\).