## 6 Simulation Test and Results Analysis
### Experimental Test of IBOA
#### 6.1.2 Results for Benchmark Test Functions  
In order to show the effectiveness of the IBOA, the commonly used test functions of CEC217 were used in the experiments. The results are compared with those obtained from WOA, GA, BOA, and HPSOBOA optimization algorithms.  
For each test function shown in Tables 1-3, each optimization algorithm was run 30 times, starting from randomly generated populations. Figure 5 shows the competitiveness of different algorithms for three test functions.  
Figure 4: Population distribution comparison map. (**a**) Cirlce mapping population distribution; (**b**) Random population distribution.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Formula of Function** & **Dim** & **Range** & \(f_{min}\) \\ \hline \(F_{1}(x)=\sum_{i=1}^{N}x_{1}^{2}\) & 30 & \([-100,100]\) & 0 \\ \(F_{2}(x)=\sum_{i=1}^{N}[x_{i}]+\prod_{i=1}^{N}[x_{i}]\) & 30 & \([-10,10]\) & 0 \\ \(F_{3}(x)=\sum_{i=1}^{N}\left(\sum_{i=1}^{N}x_{i}^{2}\right)\) & 30 & \([-100,100]\) & 0 \\ \(F_{4}(x)=\max_{i}\{[x_{i}],1\leq i\leq N\}\) & 30 & \([-100,100]\) & 0 \\ \(F_{5}(x)=\sum_{i=1}^{N-1}\left[100(x_{i+1}-x_{i}^{2})+(x_{i}-1)^{2}\right]\) & 30 & \([-30,30]\) & 0 \\ \(F_{6}(x)=\sum_{i=1}^{N-1}[(x_{i}+0.5)]^{2}\) & 30 & \([-100,100]\) & 0 \\ \(F_{7}(x)=\sum_{i=1}^{N}x_{i}^{4}+\text{random}[0,1]\) & 30 & \([-1.28,1.28]\) & 0 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Unimodal benchmark test functions.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Formula of Function** & **Dim** & **Range** & \(f_{min}\) \\ \hline \(F_{8}(x)=\sum_{i=1}^{N}-x_{i}\sin\left(\sqrt{|x_{i}|}\right)\) & 30 & \([-500,500]\) & \(-2094.9\) \\ \(F_{9}(x)=\sum_{i=1}^{N}\left[x_{i}^{2}-10\cos(2\pi x_{i})+10\right]\) & 30 & \([-5.12,5.12]\) & 0 \\ \(F_{10}(x)=-20\exp\left(-0.2\sqrt{\frac{1}{N}\sum_{i=1}^{N}x_{i}^{2}}\right)-\) & 30 & \([-32,32]\) & 0 \\ \(\exp\left(\frac{1}{N}\sum_{i=1}^{N}\cos(2\pi x_{i})\right)+20+e\) & & & \\ \(F_{11}(x)=\frac{\pi_{100}}{\pi_{100}}\sum_{i=1}^{N}x_{i}^{2}-\prod_{i=1}^{N} \cos\left(\frac{x_{i}}{\sqrt{i}}\right)+1\) & 30 & \([-600,600]\) & 0 \\ \(F_{12}(x)=\frac{\pi_{10}}{N}\left\{10\sin(\pi y_{1})+\left(y_{n}+1\right) \right\}+\) & 30 & \([-50,50]\) & 0 \\ \(\sum_{i=1}^{N-1}(y_{i}-1)^{2}\Big{[}1+10\sin(\pi y_{i+1})+\left(y_{n}+1\right) ^{2}\Big{]}\Big{\}}\) & 30 & \([-50,50]\) & 0 \\ \(y_{i}^{N-1}=1+\frac{\pi_{i+1}}{2\pi_{10}}\Big{\}}\) & & & \\ \(u(x_{i},10,100,4)\) & & & \\ \(\left\{\begin{array}{ll}k(x_{i}-a)^{m}&x_{i}>a\\ 0&-a<x_{i}<a\\ k(x_{i}-a)^{m}&x_{i}<-a\end{array}\right.\) & & \\ \(F_{13}(x)=\)0.1\(\Big{\{}10\sin^{2}(3\pi y_{1})+\Big{\}}\) & & & \\ \(\sum_{i=1}^{N}(y_{i}-1)^{2}\Big{[}1+\sin^{2}(3\pi y_{1}+1)\Big{]}+\) & 30 & \([-50,50]\) & 0 \\ \((y_{n}+1)^{2}\Big{[}1+\sin^{2}(2\pi y_{n})\Big{]}\Big{\}}\) & & & \\ \(\sum_{i=1}^{N-1}u(x_{i},5,100,4)\) & & & \\ \hline \hline \end{tabular}
\end{table}
Table 2: Multimodal benchmark test functions.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Formula of Function** & **Dim** & **Range** & \(f_{min}\) \\ \hline \(F_{14}(x)=\Big{(}\frac{1}{500}+\sum_{j=1}^{25}\frac{1}{j+\sum_{j=1}^{25}\left( \frac{x_{i}-a_{j}}{a_{j}}\right)^{6}}\Big{)}^{-1}\) & 2 & \([-65,65]\) & 1 \\ \(F_{5}(x)=\sum_{i=1}^{41}\Big{[}a_{i}-\frac{x_{i}\left(\frac{x_{i}^{2}+bx_{i}}{b _{i}+bx_{i}}\right)^{2}}{b_{i}+bFigure 5: Comparison of convergence curves of IBOA and other algorithms obtained in some of the benchmark problems.  
The comparison of the convergence curves of different algorithms reveals that the IBOA is very competitive compared with other metaheuristic techniques for solving optimization problems.