# A Novel Binary QUasi-Affine TRansformation Evolutionary (QUATRE) Algorithm
## 4 Experimental Results and Analysis  
### Benchmark Function  
In this section, the proposed BQUATRE1 and BQUATRE2 are examined by 23 benchmark functions. The mathematical formulas and properties of these functions are described in Tables 10-12; especially, D means the dimension of function and \(f_{min}\) represents the optimum. In detail, Table 10 contains the unimodal functions (denoted as f1-f7), Table 11 showsthe details of common multimodal functions (denoted as f8-f13) and Table 12 displays the details of multimodal functions in low dimension (denoted as f14-f23). Unimodal functions have only a global optimal solution and no local optimal solution, which can verify whether the algorithms can find the global optimal solution within the finite population size and iterations. Multimodal functions have many local optima, which can verify whether the algorithms can fall into the local optimal solutions or not. Since the function dimension is too low, the algorithms are prone to precocity and fall into local optimum. Therefore, the multimodal functions in low dimension can verify the convergence results under more stringent conditions. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Name** & **Function** & **Search Space** & **D** & \(f_{min}\) \\ \hline [PERSON] & \(f_{1}(x)=\sum_{i=1}^{n}x_{i}^{2}\) & [\(-\)100,100] & 30 & 0 \\ [PERSON]’s function 2.21 & \(f_{2}(x)=\sum_{i=1}^{n}|x_{i}+\prod_{i=1}^{n}|x_{i}|\) & [\(-\)10,10] & 30 & 0 \\ [PERSON]’s function 1.2 & \(f_{3}(x)=\sum_{i=1}^{n}(\sum_{i=1}^{n}x_{i})^{2}\) & [\(-\)100,100] & 30 & 0 \\ [PERSON]’s function 2.22 & \(f_{4}(x)=\max_{i}\{|x_{i}|,1\leq i\leq n\}\) & [\(-\)100,100] & 30 & 0 \\ Rosenbroke & \(f_{5}(x)=\sum_{i=1}^{n}[100(x_{i+1}-x_{i}^{2})^{2}+(x_{i}-1)^{2}]\) & [\(-\)30,30] & 30 & 0 \\ Step & \(f_{6}(x)=\sum_{i=1}^{n}([x_{i}+0.5])^{2}\) & [\(-\)100,100] & 30 & 0 \\ [PERSON]’s noisy & \(f_{7}(x)=\sum_{i=1}^{n}ix_{i}^{4}+rand[0,1)\) & [\(-\)1.28,1.28] & 30 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Unimodal benchmark functions. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Name** & **Function** & **Search Space** & **D** & \(f_{min}\) \\ \hline Schwefel & \(f_{8}(x)=\sum_{i=1}^{n}-x_{i}\sin(\sqrt{|x_{i}|})\) & [\(-\)500,500] & 30 & \(-\)12,569 \\ Rastringin & \(f_{9}(x)=\sum_{i=1}^{n}[x_{i}^{2}-10\cos(2\pi x_{i})+10]\) & [\(-\)5.12,5.12] & 30 & 0 \\ Ackley & \(f_{10}(x)=-20\exp(-0.2\sqrt{\frac{1}{n}\sum_{i=1}^{n}x_{i}^{2}})\) & [\(-\)32,32] & 30 & 0 \\ Griewank & \(-\exp(\frac{1}{n}\sum_{i=1}^{n}\cos(2\pi x_{i}))+20+e\) & & & \\ Griewank & \(f_{11}(x)=\frac{1}{2000}\sum_{i=1}^{n}x_{i}^{2}-\prod_{i=1}^{n}\cos(\frac{x_{i} }{\sqrt{i}})+1\) & [\(-\)600,600] & 30 & 0 \\  & \(f_{12}=\frac{\pi}{n}\{10\sin(\pi y_{1})+\sum_{i=1}^{n}(y_{i}-1)^{2}\) & & & \\  & \([1+10\sin^{2}(\pi y_{i+1})]+(y_{n}-1)^{2}\}\) & & & \\  & \(+\sum_{i=1}^{n}u(x_{i},0,100,4)\) & & & \\ Generalized penalized 1 & \(y_{i}=1+\frac{x_{i}+1}{4}\) & [\(-\)50,50] & 30 & 0 \\  & \(u(x_{i},a,k,m)=\) & & & \\  & & & \\  & & & \\  & & & \\ Generalized penalized 2 & \(+\sum_{i=1}^{n}(x_{i}-1)^{2}[1+\sin^{2}(3\pi x_{i}+1)]\) & & & \\  & & & \\  & & & \\  & & & \\  & & & \\  & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Common multimodal benchmark functions. Two approaches of BQUATRE algorithm with four families of transfer functions are examined in the experiment. Due to space limitation, we choose the second function of these families of transfer functions as an example, that is \(S_{2}\), \(V_{2}\), \(U_{2}\), \(Z_{2}\), \(S_{2i}\), \(V_{2i}\), \(U_{2i}\) and \(Z_{2i}\).