## 2 Bionic Social Learning Strategy Pigeon-Inspired Optimization
### Convergence Proof of the Bionic Social Learning Strategy Pigeon-Inspired Optimization  
In the BSLPIO, \(R\) adjusts the inertia weight of velocity, while \(rand\cdot(\mathbf{X}_{mean}(t)-\mathbf{X}_{i}(t))\) and \(rand\cdot(\mathbf{X}_{center_{i}}(t)-\mathbf{X}_{i}(t))\) represent bionic social learning behaviors.  
Exploration is essential in the initial stage to perform the global calculations and locate the optimum regions. To achieve a smooth transition between the two stages, the sigmoid functions \(\Big{(}1-1\Big{/}\Big{(}1+\mathrm{e}^{-(t-T_{max_{1}})\cdot\alpha}\Big{)} \Big{)}\) and \(\Big{(}1/\Big{(}1+\mathrm{e}^{-(t-T_{max_{1}})\cdot\alpha}\Big{)}\Big{)}\) are designed with additional \(\alpha\), and thus, it is necessary to analyze its influence. Figure 3a,b present the cases of \(\alpha=0.1,0.4,0.7,1\) with \(T_{max_{1}}=100\), \(T_{max_{1}}+T_{max_{2}}=200\). It can be seen that the function values \(\Big{(}1-1\Big{/}\Big{(}1+\mathrm{e}^{-(t-T_{max_{1}})\cdot\alpha}\Big{)} \Big{)}\) decrease with \(t\). The curves become steeper as \(\alpha\) increases. When \(t\leq T_{max_{1}}\), the values are nearly equal to 1. When \(T_{max_{1}}<t\leq T_{max_{1}}+T_{max_{2}}\), the values are approximately equal to 0. As to the function \(\Big{(}1/\Big{(}1+\mathrm{e}^{-(t-T_{max_{1}})\cdot\alpha}\Big{)}\Big{)}\), when \(\alpha\)= 1, it maintains the relative independence of the two stages while ensuring a smooth transition. Therefore, the speed and position updating can be rewritten as follows:  
\[\begin{split}\mathbf{V}_{i}(t+1)=\Big{(}1-1/\Big{(}1+\mathrm{e}^ {-(t-T_{max_{1}})}\Big{)}\Big{)}\big{(}\mathbf{V}_{i}(t)\cdot e^{-R\cdot t}+ rand\cdot(\mathbf{X}_{mean}(t)-\mathbf{X}_{i}(t))\big{)}+\\ \Big{(}1/\Big{(}1+\mathrm{e}^{-(t-T_{max_{1}})}\Big{)}\Big{)} \cdot rand\cdot(\mathbf{X}_{center_{i}}(t)-\mathbf{X}_{i}(t))\end{split} \tag{10}\]  
\[\mathbf{X}_{i}(t+1)=\mathbf{X}_{i}(t)+\mathbf{V}_{i}(t+1) \tag{11}\]  
Then, Formulas (10) and (11) can be rewritten into matrix form:  
\[\left(\begin{array}{c}\mathbf{V}_{i}(t+1)\\ \mathbf{X}_{i}(t+1)\end{array}\right)=\mathbf{A}\cdot\left(\begin{array}{c} \mathbf{V}_{i}(t)\\ \mathbf{X}_{i}(t)\end{array}\right)+\mathbf{B}\cdot\left(\begin{array}{c} \mathbf{X}_{mean}\\ \mathbf{X}_{center_{i}}\end{array}\right) \tag{12}\]  
\[\mathbf{A}=\begin{pmatrix}w_{1}&-(w_{2}+w_{3})\\ w_{1}&1-(w_{2}+w_{3})\end{pmatrix},\mathbf{B}=\begin{pmatrix}w_{2}\\ w_{3}\end{pmatrix} \tag{13}\]  
where \(w_{1}=\Big{(}1-1/\Big{(}1+\mathrm{e}^{-(t-T_{max_{1}})}\Big{)}\Big{)}\cdot e^ {-R\cdot t}\), \(w_{2}=\Big{(}1-1/\Big{(}1+\mathrm{e}^{-(t-T_{max_{1}})}\Big{)}\Big{)}\cdot rand\), \(w_{3}=\Big{(}1/\Big{(}1+\mathrm{e}^{-(t-T_{max_{2}})}\Big{)}\Big{)}\cdot rand\).  
Figure 3: The designed sigmoid functions: (**a**) the sigmoid function 1; (**b**) the sigmoid function 2.  
The characteristic polynomial of the matrix \(\mathbf{A}\) is as follows:  
\[|\lambda\mathbf{I}-\mathbf{A}|=\lambda^{2}-(w_{2}+w_{3}-1-w_{1})\lambda+w_{1} \tag{14}\]  
where \(\lambda_{1}\) and \(\lambda_{2}\) are the roots.  
The spectral radius of the matrix \(\mathbf{A}\) is as follows:  
\[\rho(\mathbf{A})=\max_{1\leq i\leq 2}|\lambda_{i}|=\{|\lambda_{1}|,|\lambda_{2}|\} \tag{15}\]  
When \(-1<w_{1}\leq 0\), \(\rho(A)\) is expressed as follows:  
\[\rho(\mathbf{A})=\left\{\begin{array}{l}\frac{\gamma_{1}+\gamma_{2}}{2},0<(w _{2}+w_{3})<1+w_{1}\\ \frac{-\gamma_{1}+\gamma_{2}}{2},1+w_{1}<(w_{2}+w_{3})<2+2w_{1}\end{array}\right. \tag{16}\]  
where \(\gamma_{1}=-(w_{2}+w_{3})+1+w_{1}\), \(\gamma_{2}=\sqrt{((w_{2}+w_{3})-1-w)^{2}-4w_{1}}\).  
Therefore, the parameters' ranges satisfied \(\rho(\mathbf{A})<1\) are given by  
\[\left\{\begin{array}{l}-1<w_{1}<1\\ 0<(w_{2}+w_{3})<2+2w_{1}\end{array}\right. \tag{17}\]  
For the BSLSPIO, we have \(w_{1},w_{2},w_{3}\in(0,1)\). Therefore, the proposed algorithm satisfies the convergence conditions.