## 3 Results  
### Objective Function Values  
In all 12 test examples, the minimum (target) value of the objective function is zero, as shown in Table 3. The results of the PROS algorithm in terms of minimum objective value achieved, are presented in Table 4 (as the average of 10 runs). Figure 3: Time needed for the calculation of each objective function 100,000 times, for a random vector of 5, 10, 30 or 50 variables (\(D\) = 5, 10, 30, 50). The y-axis is in logarithmic scale. The PROS algorithm shows a very good performance in the test functions examined. It clearly outperforms PRS which is very poor and although PROS is so simple in its formulation, the algorithm gives very competitive and even better results in comparison to the other established algorithms, i.e., GA, PSO and DE. In many cases, PROS outperforms some, or all, of the other algorithms. Surprisingly, the algorithm shows very good performance (compared to the others) in difficult cases with a large number of dimensions (\(D=30\) or \(D=50\)). As expected, PRS exhibits the poorest performance with very large values of the objective function in some cases. Figure 4 shows the best results (final values of the objective values) for each algorithm, for the different cases (as averages of 10 runs). PRS has been excluded from this comparison because its final objective function values are very large, and it would distort the diagram significantly. For better presentation of the results, the y-axis has been limited to the value of 1 in all cases. Since the target (optimum) value of each of the 12 functions used is zero, the lower bar height is better in this comparison. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**ID**} & \multicolumn{2}{c}{**Function**} & \multirow{2}{*}{**GA**} & \multirow{2}{*}{**PSO**} & \multirow{2}{*}{**DE**} & \multirow{2}{*}{**PRS**} & \multirow{2}{*}{**PROS**} \\  & & & & & & & \\ \hline F01 & Sphere & 6.93E-06 & 4.41E-11 & 1.02E-01 & **4.83E+03** & 9.56E-02 \\ F02 & Ellipsoid & 5.89E-05 & 3.83E-10 & 4.03E-01 & **1.83E+04** & 6.72E-01 \\ F03 & Sum of Dif. & 4.22E-08 & 7.75E-23 & 1.57E-06 & **2.04E+03** & 5.58E-05 \\ F04 & Quintic & 2.42E-02 & 3.52E-05 & **1.04E+01** & **4.95E+04** & 6.61E-01 \\ F05 & Drop- & 3.45E-01 & 1.24E-01 & 1.73E-01 & 7.85E-01 & 5.48E-01 \\ F06 & Weierstrass & 1.24E-01 & 9.90E-05 & 8.30E-01 & **1.03E+01** & 3.62E-01 \\ F07 & Alpine 1 & 1.10E-03 & 4.56E-07 & **1.68E+00** & **7.33E+00** & 3.16E-03 \\ F08 & Ackley’s & 1.94E-03 & 1.98E-06 & 2.66E-01 & **1.70E+01** & 1.72E-01 \\ F09 & Griewank’s & 1.06E-02 & 1.48E-01 & 4.83E-01 & **2.11E+00** & 4.13E-02 \\ F10 & Rastrigin’s & 1.96E-04 & **8.12E+00** & **3.61E+01** & **6.62E+01** & 4.94E-02 \\ F11 & HappyCat & 7.92E-01 & 1.86E-01 & 3.98E-01 & **1.14E+01** & 4.25E-01 \\ F12 & HGBat & 8.00E-01 & 3.00E-01 & 3.09E-01 & **1.02E+02** & 4.61E-01 \\ \hline \hline \end{tabular}  
* Average value (in 10 runs) of the obj. function at the end of the optimization process, for each algorithm.  
\end{table} 
Table 6: Best results * for all 5 optimizers and all 12 objective functions, for 10 dimensions (\(D=10\), averages of 10 runs). The target (optimum) value is zero in all cases considered.