## 6 Experimental Results
### Benchmark Functions  
VLE was tested using 15 mathematical benchmark functions frequently used by many researchers, Equations (19)-(33) [20; 24; 30; 34; 42], and 6 composite benchmark functions selected from the Technical Report of the CEC 2017 Special Session [43], as described in Table 6. The first set of benchmark functions includes four unimodal functions, Equations (19)-(22), five multimodal functions, Equations (23)-(27), and six multimodal functions with fix dimensions, Equations (28)-(33). Functions \(f_{1}\) to \(f_{9}\), i.e., Equations (19)-(27), are high-dimensional problems. Functions \(f_{5}\) to \(f_{9}\), i.e., Equations (23)-(27), are a very difficult group of problems for optimization algorithms. In these problems, the number of local minima increases exponentially as the number of dimensions increases [20]. This set of multimodal functions is very important because it reflects the ability of startup from local optima and continues the search in another place in the search space. The number of dimensions is \(n=30\) in Equations (19)-(27). The minimum values of all these functions and the corresponding solutions are given in Table 3. Figures 13-15 show 3D views of the first set of benchmark functions utilized in our experiments. **Unimodal test functions**:  
Sphere Function:  
\[f_{1}(X)=\sum_{i=1}^{n}x_{i}^{2} \tag{19}\]  
Schwefel's Function No. 222:  
\[f_{2}(X)=\sum_{i=1}^{n}|x_{i}|+\prod_{i=1}^{n}|x_{i}| \tag{20}\]  
Schwefel's Function No. 1.2:  
\[f_{3}(X)=\sum_{i=1}^{n}\left(\sum_{j=1}^{i}x_{j}\right)^{2} \tag{21}\]  
Generalized Rosenbrock's Function:  
\[f_{4}(X)=\sum_{i=1}^{n-1}\left[100(x_{i+1}-x_{i}^{2})^{2}+(x_{i}-1)^{2}\right] \tag{22}\]  
Figure 13: _Cont._ \begin{table}
\begin{tabular}{c c c c} \hline \hline
**BenFun** & **SeaSub** & **Opt** & **Sol** \\ \hline \(f_{1}(X)\) & \([-100,100]^{30}\) & \(0\) & \([0]^{30}\) \\ \(f_{2}(X)\) & \([-10,10]^{30}\) & \(0\) & \([0]^{30}\) \\ \(f_{3}(X)\) & \([-100,100]^{30}\) & \(0\) & \([0]^{30}\) \\ \(f_{4}(X)\) & \([-30,30]^{30}\) & \(0\) & \([1]^{30}\) \\ \(f_{5}(X)\) & \([-500,500]^{30}\) & \(-12,569.487\) & \([420.9687]^{30}\) \\ \(f_{6}(X)\) & \([-5.12,5.12]^{30}\) & \(0\) & \([0]^{30}\) \\ \(f_{7}(X)\) & \([-32,32]^{30}\) & \(0\) & \([0]^{30}\) \\ \(f_{8}(X)\) & \([-600,600]^{30}\) & \(0\) & \([0]^{30}\) \\ \(f_{9}(X)\) & \([-50,50]^{30}\) & \(0\) & \([1]^{30}\) \\ \(f_{10}(X)\) & \([-655.36,65.536]^{2}\) & \(1\) & \([-32]^{2}\) \\ \(f_{11}(X)\) & \([-5,5]^{2}\) & \(-1.0316285\) & \((0.08983,-0.7126)\) and \((-0.08983,0.7126)\) \\ \(f_{12}(X)\) & \([-5,10]\) for \(x_{1}\) and \(0.397887\) & \((-3.142,12.275)\), \((3.142,2.275)\) and \\ \([0,15]\) for \(x_{2}\) & & \((9.425,2.425)\) \\ \(f_{13}(X)\) & \([-2,2]^{2}\) & \(3\) & \((0,-1)\) \\ \(f_{14}(X)\) & \([0,1]^{3}\) & \(-3.86\) & \((0.114,0.556,0.852)\) \\ \(f_{15}(X)\) & \([0,1]^{6}\) & \(-3.32\) & \((0.201,0.150,0.477,0.275,0.311,0.657)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Optimum values reported for the benchmark functions in the literature, with their corresponding solutions and search subsets. **Multimodal test functions**:  
Generalized Schwefel's Function No. 2.26:  
\[f_{5}(X)=-{\sum_{i=1}^{m}}\left(x_{i}\sin\left(\sqrt{|x_{i}|}\right)\right) \tag{23}\]  
Generalized Rastrigin's Function:  
\[f_{6}(X)={\sum_{i=1}^{m}}\left[x_{i}^{2}-10\cos\left(2\pi x_{i} \right)+10\right] \tag{24}\]  
Ackley's Function:  
\[f_{7}(X)=-20\exp\left(-0.2\sqrt{\frac{1}{n}{\sum_{i=1}^{n}}x_{i}^{2}}\right)- \exp\left(\frac{1}{n}{\sum_{i=1}^{n}}\cos\left(2\pi x_{i}\right) \right)+20+e \tag{25}\]  
Generalized Griewank's Function:  
\[f_{8}(X)=\frac{1}{4000}{\sum_{i=1}^{n}}x_{i}^{2}-{\prod_{i=1}^{n}}\cos\left( \frac{x_{i}}{\sqrt{i}}\right)+1 \tag{26}\]  
Generalized Penalized Function:  
\[ \begin{array}{l}f_{9}(X)=\frac{\pi}{n}\left\{10\sin^{2}\left(\pi y_{1} \right)+{\sum_{i=1}^{n-1}}\left(y_{i}-1\right)^{2}\left[1+10\sin^{2} \left(\pi y_{i+1}\right)\right]+\left(y_{n}-1\right)^{2}\right\}\\ \hskip 142.26378 pt+\sum_{i=1}^{n}u\left(x_{i},10,100,4\right)\end{array}