## 4 Experiments
### Experiment I: Basic Benchmark Problems  
Figure 4 shows the experimental results of the mean error for 100 independent runs of each of the algorithms on each benchmark problem for dimension \(D=5\). The x-axes are the number of objective function evaluations (in log scale) and the y-axes are the average error of the algorithm after certain objective function evaluations. As shown in the figure, QROS converged faster than the other five algorithms except on \(f_{3}\) and \(f_{5}\) where PSO converged the fastest. TROS and PROS had similar trends as QROS on the benchmark functions: they converged faster than GA, PSO and DE on most benchmark functions \begin{table}
\begin{tabular}{c c c c} \hline \hline
**No.** & **Function** & **Formulation and Global Optimum** & **Search Space** \\ \hline \(f_{6}\) & Alpine \(1\) & \(\sum_{i=1}^{D}|x_{i}\sin(x_{i})+0.1x_{i}|\) & \([-10,10]^{D}\) \\  & & \(\mathbf{x}^{*}=(0,0,\ldots,0),f(\mathbf{x}^{*})=0\) & \\ \hline \(f_{7}\) & Rastrigin & \(10D+\sum_{i=1}^{D}[x_{i}^{2}-10\cos(2\pi x_{i})]\) & \([-5.12,5.12]^{D}\) \\  & & \(\mathbf{x}^{*}=(0,0,\ldots,0),f(\mathbf{x}^{*})=0\) & \\ \hline \(f_{8}\) & Ackley & \(20+\exp(1)-20\exp(-0.2\sqrt{\frac{1}{D}\sum_{i=1}^{D}x_{i}^{2}})\) & \([-32.768,32.768]^{D}\) \\  & & \(-\exp(\frac{1}{D}\sum_{i=1}^{D}\cos(2\pi x_{i}))\) & \\  & & \(\mathbf{x}^{*}=(0,0,\ldots,0),f(\mathbf{x}^{*})=0\) & \\ \hline \(f_{9}\) & Griewank & \(\sum_{i=1}^{D}\frac{x_{i}^{2}}{4000}-\prod_{i=1}^{D}\cos\left(\frac{x_{i}}{ \sqrt{i}}\right)+1\) & \([-600,600]^{D}\) \\  & & \(\mathbf{x}^{*}=(0,0,\ldots,0),f(\mathbf{x}^{*})=0\) & \\ \hline \(f_{10}\) & HGBat & \(|(\sum_{i=1}^{D}x_{i}^{2})^{2}-(\sum_{i=1}^{D}x_{i})^{2}|^{1/2}+\) & \([-15,15]^{D}\) \\  & & \((0.5\sum_{i=1}^{D}x_{i}^{2}+\sum_{i=1}^{D}x_{i})/D+0.5\) & \\  & & \(\mathbf{x}^{*}=(-1,-1,\ldots,-1),f(\mathbf{x}^{*})=0\) & \\ \hline \(f_{11}\) & HappyCat & \(|\sum_{i=1}^{D}x_{i}^{2}-D|^{1/4}+\) & \([-20,20]^{D}\) \\  & & \((0.5\sum_{i=1}^{D}x_{i}^{2}+\sum_{i=1}^{D}x_{i})/D+0.5\) & \\  & & \(\mathbf{x}^{*}=(-1,-1,\ldots,-1),f(\mathbf{x}^{*})=0\) & \\ \hline \(f_{12}\) & Weierstrass & \(\sum_{i=1}^{D}[\sum_{i=0}^{20}0.5^{k}\cos(2\pi 3^{k}(x_{i}+0.5))\) & \([-0.5,0.5]^{D}\) \\  & & \(-D\sum_{i=0}^{D}0.5^{k}\cos(\pi 3^{k})]\) & \\  & & \(\mathbf{x}^{*}=(0,0,\ldots,0),f(\mathbf{x}^{*})=0\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: _Cont._ \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Settings** & \(\boldsymbol{D=5}\) & \(\boldsymbol{D=10}\) & \(\boldsymbol{D=50}\) \\ \hline Population Size & \(10D\) & \(50\) & \(100\) & \(500\) \\ \hline Max. No. of Generations & \((20D-50)\) & \(50\) & \(150\) & \(950\) \\ \hline Max. No. of Objective & \(10D(20D-50)\) & \(2500\) & \(15,000\) & \(475,000\) \\ Function Evaluations & & & \\ \hline No. of Runs & \(100\) & \(100\) & \(30\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Settings of the numerical experiments. Table 3 shows the mean of the errors of the final solution vectors returned by PROS, TROS and QROS for 100 independent runs of on each benchmark problem for dimension \(D=5\). The mean errors represent the medium to long term performance of the algorithms. The best results (the smallest mean final errors) for each benchmark problem are highlighted in bold font. The corresponding standard deviations are placed next with parenthesis. Owing to limited space, the results of GA, PSO and DE are not shown here, as they have been reported in [15] already. As seen from the table, QROS had the smallest mean final errors on most of the benchmark problems. Similar results can be seen from Tables 4 and 5 which show the means of the final errors for dimension \(D=10\) and \(D=50\), respectively. In general, QROS and TROS are more efficient in reducing the mean final errors on most benchmark problems when compared with PROS.