# Combining Parallel Stochastic Methods and Mixed Termination Rules in Optimization
## 2 Adopted Algorithms
### The Method Multistart
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Problem** & **Best** & **Mean** & **DoubleBox** & **All** \\ \hline EXP16 & 5588 & 27,387 & 7552 & 5339 \\ GKLS250 & 5227 & 2641 & 8379 & 2641 \\ GKLS350 & 5624 & 3298 & 17,437 & 3206 \\ GRIEWANK2 & 8027 & 10,458 & 14,756 & 6915 \\ GRIIEWANK10 & 9664 & 37,839 & 17,312 & 9539 \\ POTENTIAL3 & 5278 & 24,823 & 13,871 & 5256 \\ POTENTIAL5 & 8225 & 32,439 & 20,828 & 7742 \\ POTENTIAL6 & 8467 & 34,946 & 19,169 & 8197 \\ POTENTIAL10 & 10,330 & 41,300 & 26,308 & 10,643 \\ HANSEN & 5219 & 23,050 & 14,245 & 4263 \\ HARTMAN3 & 4613 & 17,966 & 7333 & 4566 \\ HARTMAN6 & 5734 & 17,109 & 9354 & 5550 \\ RASTRIGIN & 5912 & 17,240 & 8987 & 5999 \\ ROSENBROCK8 & 8503 & 34,429 & 20,980 & 8051 \\ ROSENBROCK16 & 10,052 & 41,335 & 26,156 & 8307 \\ SHEKEL5 & 6064 & 25,800 & 19,877 & 5933 \\ SHEKEL7 & 5597 & 26,168 & 20,274 & 5677 \\ SHEKEL10 & 6037 & 26,439 & 22,960 & 5100 \\ SINU4 & 5751 & 24,620 & 12,511 & 4776 \\ SINU8 & 6340 & 25,477 & 18,741 & 4980 \\ TEST2N4 & 5848 & 24,947 & 11,640 & 5727 \\ TEST2N9 & 7303 & 26,571 & 16,165 & 6288 \\ TEST30N3 & 3169 & 8395 & 8134 & 2869 \\ TEST30N4 & 2841 & 10,381 & 7853 & 2714 \\
**Total** & **214,552** & **677,557** & **474,733** & **186,197** \\ \hline \hline \end{tabular}
\end{table} 
Table 3: _Cont._

Figure 2: Statistical comparison of function calls with different termination rules of differential evolutions. two techniques that seem to be the most decisive for terminating each function are the Best and the DoubleBox methods, and in fact the Best method appears to significantly outperform the DoubleBox termination scheme on average. Subsequently, the same experiment was conducted for the PSO method and the results are shown in Table 5 and the statistical comparison is outlined graphically in Figure 3. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Problem** & **Mean** & **Best** & **DoubleBox** \\ \hline BF1 & 0.00\% & 36.67\% & 63.33\% \\ BF2 & 6.67\% & 33.33\% & 60.00\% \\ BRANIN & 0.00\% & 6.67\% & 93.33\% \\ CAMEL & 10.00\% & 76.67\% & 13.33\% \\ CIGAR10 & 0.00\% & 20.00\% & 80.00\% \\ CM4 & 0.00\% & 90.00\% & 10.00\% \\ DISCUS10 & 0.00\% & 6.67\% & 93.33\% \\ EASOM & 0.00\% & 0.00\% & 100.00\% \\ ELP10 & 0.00\% & 0.00\% & 100.00\% \\ EXP4 & 0.00\% & 86.67\% & 13.33\% \\ EXP16 & 0.00\% & 76.67\% & 23.33\% \\ GKLS250 & 6.67\% & 73.33\% & 20.00\% \\ GKLS350 & 20.00\% & 80.00\% & 0.00\% \\ GRIEWANK2 & 3.33\% & 26.67\% & 70.00\% \\ GRIEWANK10 & 0.00\% & 10.00\% & 90.00\% \\ POTENTIAL3 & 3.33\% & 66.67\% & 30.00\% \\ POTENTIAL5 & 10.00\% & 86.67\% & 3.33\% \\ POTENTIAL6 & 0.00\% & 90.00\% & 10.00\% \\ PONTENTIAL10 & 13.33\% & 86.67\% & 0.00\% \\ HANSEN & 16.67\% & 50.00\% & 33.33\% \\ HARTMAN3 & 0.00\% & 70.00\% & 30.00\% \\ HARTMAN6 & 3.33\% & 76.67\% & 20.00\% \\ [PERSON] & 0.00\% & 83.33\% & 16.67\% \\ ROSENBROCK8 & 0.00\% & 13.33\% & 86.67\% \\ ROSENBROCK16 & 0.00\% & 13.33\% & 86.67\% \\ SHEKEL5 & 3.33\% & 93.33\% & 3.33\% \\ SHEKEL7 & 10.00\% & 90.00\% & 0.00\% \\ SHEKEL10 & 6.67\% & 93.33\% & 0.00\% \\ SINU4 & 6.67\% & 80.00\% & 13.33\% \\ SINU8 & 0.00\% & 100.00\% & 0.00\% \\ TEST2N4 & 0.00\% & 76.67\% & 23.33\% \\ TEST2N9 & 0.00\% & 100.00\% & 0.00\% \\ TEST30N3 & 0.00\% & 33.33\% & 66.67\% \\ TEST30N4 & 0.00\% & 40.00\% & 60.00\% \\
**AVERAGE** & **3.53\%** & **57.84\%** & **38.63\%** \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Percentage of the rule which is responsible for termination in the mixed stopping rule. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Problem** & **Best** & **Mean** & **DoubleBox** & **All** \\ \hline BF1 & 3418 & 3127 & 3122 & 3005 \\ BF2 & 3265 & 2995 & 2919 & 2889 \\ BRANIN & 2601 & 2549 & 2431 & 2417 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Average function calls for the PSO method, with different termination rules.