## 5 Numerical Experiments
### Algorithm Complexity  
In this Section, the algorithm complexity for the proposed GA-TNE+DRO is described. Table 6 shows the complexity of the four algorithms: GA, PSO, DE, and GA-TNE+DRO. Using the method in [33], the running times of algorithms are measured against T0, the running time of Algorithm 2. Under our experimental setup, \(T0\) is \(3.4\times 10^{-5}\) s.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Function**} & \multicolumn{2}{c}{**GA**} & \multicolumn{2}{c}{**PSO**} & \multicolumn{2}{c}{**DE**} & \multicolumn{2}{c}{**GA-TNE+DRO**} \\ \cline{2-10}  & \(F_{best}\) & \(F_{ave}\) & \(F_{best}\) & \(F_{ave}\) & \(F_{best}\) & \(F_{best}\) & \(F_{ave}\) & \(F_{best}\) & \(F_{ave}\) \\ \hline \(f_{1}(\mathbf{x})\) & \(8.43\times 10^{-8}\) & \(4.91\times 10^{-7}\) & \(3.13\times 10^{-10}\) & \(3.67\times 10^{-8}\) & \(4.13\times 10^{-9}\) & \(3.18\times 10^{-8}\) & \(\mathbf{1.13\times 10^{-11}}\) & \(\mathbf{1.16\times 10^{-10}}\) \\ \(f_{2}(\mathbf{x})\) & \(3.48\times 10^{-7}\) & \(6.12\times 10^{-6}\) & \(5.62\times 10^{-9}\) & \(4.12\times 10^{-8}\) & \(7.52\times 10^{-10}\) & \(7.01\times 10^{-9}\) & \(\mathbf{7.85\times 10^{-11}}\) & \(\mathbf{8.61\times 10^{-10}}\) \\ \(f_{3}(\mathbf{x})\) & \(1.86\times 10^{-8}\) & \(3.65\times 10^{-6}\) & \(1.58\times 10^{-10}\) & \(4.28\times 10^{-9}\) & \(3.04\times 10^{-10}\) & \(3.17\times 10^{-9}\) & \(\mathbf{2.47\times 10^{-12}}\) & \(\mathbf{3.45\times 10^{-11}}\) \\ \(f_{4}(\mathbf{x})\) & \(-20.0023\) & \(-20.0238\) & \(-19.9986\) & \(-19.9865\) & \(-19.9987\) & \(-19.9928\) & \(-\mathbf{20.0001}\) & \(-\mathbf{20.0001}\) \\ \(f_{5}(\mathbf{x})\) & \(2.36\times 10^{-5}\) & \(5.69\times 10^{-4}\) & \(3.65\times 10^{-6}\) & \(7.54\times 10^{-5}\) & \(5.97\times 10^{-6}\) & \(4.19\times 10^{-5}\) & \(\mathbf{7.98\times 10^{-7}}\) & \(\mathbf{8.67\times 10^{-6}}\) \\ \(f_{6}(\mathbf{x})\) & \(-09.6350\) & \(-09.6251\) & \(-09.96349\) & \(-09.96332\) & \(-09.96343\) & \(-09.96341\) & \(-09.96354\) & \(-09.69350\) \\ \(f_{7}(\mathbf{x})\) & \(-10.8716\) & \(-10.8689\) & \(-10.8718\) & \(-10.8702\) & \(-10.8718\) & \(-10.8703\) & \(-\mathbf{10.8721}\) & \(-\mathbf{10.8714}\) \\ \(f_{8}(\mathbf{x})\) & \(4.55\times 10^{-8}\) & \(2.36\times 10^{-6}\) & \(3.68\times 10^{-9}\) & \(6.39\times 10^{-7}\) & \(\mathbf{8.26\times 10^{-10}}\) & \(\mathbf{8.69\times 10^{-9}}\) & \(6.31\times 10^{-9}\) & \(9.23\times 10^{-8}\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: The optimal values of objective functions obtained by the algorithms (the best obtained results are highlighted in bold).  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Function**} & \multicolumn{2}{c}{**GA**} & \multicolumn{2}{c}{**PSO**} & \multicolumn{2}{c}{**DE**} & \multicolumn{2}{c}{**GA-TNE+DRO**} \\ \cline{2-11}  & \(G_{\max}\) & \(G_{\min}\) & \(G_{\text{ave}}\) & \(G_{\max}\) & \(G_{\min}\) & \(G_{\text{ave}}\) & \(G_{\max}\) & \(G_{\min}\) & \(G_{\text{ave}}\) & \(G_{\min}\) & \(G_{ave}\) \\ \hline \(f_{1}(\mathbf{x})\) & \(200\) & \(54\) & \(86.8\) & \(200\) & \(51\) & \(65.8\) & \(191\) & \(50\) & \(70.4\) & \(180\) & \(\mathbf{9}\) & \(\mathbf{15.6}\) \\ \(f_{2}(\mathbf{x})\) & \(200\) & \(102\) & \(120.4\) & \(In Table 6, \(T1\) is the running time for executing the benchmark function alone for 200 times. \(T2\) is the total running time for applying an algorithm to solve a benchmark function 200 times. \(\overline{T2}\) is the mean \(T2\) over 50 runs. The complexity of the algorithm is measured by \(T=(\overline{T2}-T1)/T0\). The best results in Table 6 are highlighted in bold. According to Table 6, our algorithm is much more efficient than the other algorithms.