## 4 Experiments
### Experiment II: Random Shifted Benchmark Problems  
One may notice that both TROS and QROS are in favor of objective functions with the global optimum located at the center of the search space. For example, the global optimum of \(f_{1},f_{2},f_{3},f_{5},f_{6},f_{7},f_{8},f_{9},f_{12}\) is \((0,0,\ldots,0)\). In order to test the effectiveness of TROS and QROS in a general case, another experiment was conducted. The same set of benchmark problems were used in this experiment, but the search space was shifted so that the global optimum may be located anywhere within the bounded search space. A random point \(\mathbf{o}=(o_{1},o_{2},\ldots,o_{D})\) is drawn uniformly from \(\Omega^{\prime}=[a_{1}-x_{1}^{*},b_{1}-x_{1}^{*}]\times[a_{2}-x_{2}^{*},b_{2}- x_{2}^{*}]\times\cdots\times[a_{D}-x_{D}^{*},b_{D}-x_{D}^{*}]\) for each benchmark function for each run where \(a_{i},b_{i},x_{i}^{*}\) are the lower limit, upper limit and the optimum value of the \(i\)-th decision variable in the original search space of the benchmark function being optimized and \(i\in\{1,2,\ldots,D\}\). The shifted search space becomes \(\Omega^{\prime\prime}=[a_{1}-o_{1},b_{1}-o_{1}]\times[a_{2}-o_{2},b_{2}-o_{2}] \times\cdots\times[a_{D}-o_{D},b_{D}-o_{D}]\). It has to be noted that the benchmark problems are carefully chosen in this experiment to \begin{table}
\begin{tabular}{c c c c} \hline \hline \(f\) & **PROS** & **TROS** & **QROS** \\ \hline \(f_{1}\) & 8.62e-04 (7.26e-04) & 2.38e-04 (1.85e-04) & **1.03e-04 (7.51e-05)** \\ \(f_{2}\) & 4.45e-03 (3.33e-03) & 1.34e-03 (1.23e-03) & **5.68e-04 (4.63e-04)** \\ \(f_{3}\) & 6.84e-01 (3.62e-01) & 2.52e-01 (1.35e-01) & **1.35e-01 (7.28e-02)** \\ \(f_{4}\) & **3.40e+00 (3.28e+00)** & 3.72e+00 (2.99e+00) & 4.53e+00 (2.91e+00) \\ \(f_{5}\) & 4.69e+01 (2.47e+01) & 1.40e+01 (9.29e+00) & **6.26e+00 (4.77e+00)** \\ \(f_{6}\) & 3.10e-03 (9.52e-04) & 1.70e-03 (5.30e-04) & **1.10e-03 (3.52e-04)** \\ \(f_{7}\) & 4.48e-02 (3.77e-02) & 1.24e-02 (9.46e-03) & **5.24e-03 (3.56e-03)** \\ \(f_{8}\) & 1.61e-01 (8.23e-02) & 7.13e-02 (3.00e-02) & **4.32e-02 (1.55e-02)** \\ \(f_{9}\) & 1.78e-01 (6.53e-02) & 9.15e-02 (3.68e-02) & **6.77e-02 (2.74e-02)** \\ \(f_{10}\) & 4.73e-01 (2.46e-01) & 4.35e-01 (1.97e-01) & **4.31e-01 (1.99e-01)** \\ \(f_{11}\) & 4.90e-01 (1.65e-01) & 4.30e-01 (1.49e-01) & **3.76e-01 (1.17e-01)** \\ \(f_{12}\) & 3.40e-01 (8.16e-02) & 2.20e-01 (4.97e-02) & **1.68e-01 (3.85e-02)** \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Statistical results of 100 runs on 10D benchmark functions. \begin{table}
\begin{tabular}{c c c c} \hline \hline \(f\) & **PROS** & **TROS** & **QROS** \\ \hline \(f_{1}\) & 1.12e-04 (3.30e-05) & 2.82e-05 (7.60e-06) & **1.24e-05 (4.68e-06)** \\ \(f_{2}\) & 2.86e-03 (9.68e-04) & 7.25e-04 (2.22e-04) & **3.29e-04 (1.32e-04)** \\ \(f_{3}\) & 1.34e+01 (3.13e+00) & 6.45e+00 (1.57e+00) & **3.75e+00 (8.21e-01)** \\ \(f_{4}\) & 8.06e+01 (3.80e+01) & 6.05e+01 (3.41e+01) & **5.48e+01 (2.46e+01)** \\ \(f_{5}\) & 9.66e+02 (1.63e+02) & 7.58e+02 (1.51e+02) & **6.44e+02 (1.36e+02)** \\ \(f_{6}\) & 2.46e-03 (3.48e-04) & 1.32e-03 (1.93e-04) & **8.85e-04 (1.26e-04)** \\ \(f_{7}\) & 5.81e-03 (1.72e-03) & 1.32e-03 (2.79e-04) & **6.25e-04 (1.60e-04)** \\ \(f_{8}\) & 2.06e-02 (3.28e-03) & 9.90e-03 (1.13e-03) & **6.49e-03 (9.52e-04)** \\ \(f_{9}\) & 2.63e-02 (1.51e-02) & **1.17e-02 (1.01e-02)** & 1.39e-02 (1.77e-02) \\ \(f_{10}\) & 5.99e-01 (2.48e-01) & 5.80e-01 (2.25e-01) & **5.66e-01 (2.27e-01)** \\ \(f_{11}\) & 6.51e-01 (1.28e-01) & **6.17e-01 (1.11e-01)** & 6.21e-01 (1.08e-01) \\ \(f_{12}\) & 5.34e-01 (4.80e-02) & 3.43e-01 (2.93e-02) & **2.61e-01 (2.84e-02)** \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Statistical results of 30 runs on 50D benchmark functions.