# 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 BF1 & 53,873 & 479,374 & 50,762 & 50,762 \\ BF2 & 37,670 & 239,220 & 35,711 & 35,711 \\ BRANIN & 11,063 & 11,046 & 10,521 & 10,521 \\ CAMEL & 16,083 & 153,559 & 15,255 & 15,255 \\ CIGAR10 & 14,937 & 14,937 & 14,697 & 14,697 \\ CM4 & 62,236 & 62,192 & 58,581 & 58,581 \\ DISCUS10 & 6296 & 6296 & 6056 & 6056 \\ EASOM & 5412 & 55,130 & 51,132 & 5412 \\ ELP10 & 14,757 & 14,981 & 14,517 & 14,517 \\ EXP4 & 10,022 & 54,174 & 9774 & 9774 \\ EXP16 & 10,528 & 54,680 & 10,280 & 10,280 \\ GKLS250 & 6202 & 47,948 & 5908 & 5908 \\ GKLS350 & 7560 & 42,145 & 7087 & 7087 \\ GRIEWANK2 & 19,179 & 188,429 & 17,877 & 17,877 \\ GRIEWANK10 & 65,206 & 567,153 & 61,818 & 61,818 \\ POTENTIAL3 & 17,773 & 126,632 & 17,161 & 17,161 \\ POTENTIAL5 & 31,347 & 226,873 & 30,249 & 30,249 \\ POTENTIAL6 & 35,457 & 247,144 & 33,883 & 33,883 \\ POTENTIAL10 & 44,957 & 283,230 & 43,618 & 43,618 \\ HANSEN & 18,568 & 201,543 & 17,541 & 17,541 \\ HARTMAN3 & 17,395 & 162,934 & 16,562 & 16,562 \\ HARTMAN6 & 22,010 & 179,073 & 21,015 & 21,015 \\ RASTRIGIN & 24,696 & 275,610 & 23,015 & 23,015 \\ ROSENBROCK8 & 17,850 & 17,850 & 17,610 & 17,610 \\ ROSENBROCK16 & 25,300 & 25,300 & 25,060 & 25,060 \\ SHEKEL5 & 17,725 & 151,648 & 16,972 & 16,972 \\ SHEKEL7 & 17,894 & 154,373 & 17,127 & 17,127 \\ SHEKEL10 & 17,931 & 159,825 & 17,135 & 17,135 \\ SINU4 & 16,429 & 155,634 & 15,647 & 15,647 \\ SINU8 & 19,519 & 170,167 & 18,674 & 18,674 \\ TEST2N4 & 16,486 & 154,092 & 15,806 & 15,806 \\ TEST2N9 & 22,035 & 160,651 & 20,716 & 20,716 \\ TEST30N3 & 16,934 & 161,628 & 16,145 & 16,145 \\ TEST30N4 & 16,686 & 159,753 & 15,907 & 15,907 \\
**Total** & **758,016** & **5,165,224** & **769,819** & **724,099** \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Multistart function calls with different termination rules. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Problem** & **Best** & **Mean** & **DoubleBox** & **All** \\ \hline BF1 & 3080 & 4675 & 6719 & 3080 \\ BF2 & 3083 & 4747 & 6923 & 3083 \\ BRANIN & 3217 & 3217 & 5105 & 2661 \\ CAMEL & 2750 & 3411 & 3780 & 2724 \\ CIGAR10 & 14,718 & 23,003 & 24,134 & 14,718 \\ CM4 & 3187 & 5638 & 8802 & 3187 \\ DISCUS10 & 12,946 & 22,194 & 24,134 & 12,946 \\ EASOM & 2701 & 2965 & 10,201 & 2135 \\ ELP10 & 6440 & 22,449 & 24,123 & 6440 \\ EXP4 & 2796 & 5198 & 7830 & 2796 \\ EXP16 & 4991 & 10,842 & 22,025 & 4991 \\ GKLS250 & 3072 & 3072 & 4840 & 2940 \\ GKLS350 & 4041 & 4041 & 8620 & 4041 \\ GRIEWANK2 & 3155 & 4818 & 7258 & 3155 \\ GRIEWANK10 & 8568 & 10,196 & 22,841 & 8568 \\ POTENTIAL3 & 3766 & 3766 & 13,063 & 3701 \\ POTENTIAL5 & 4082 & 4082 & 19,826 & 3645 \\ POTENTIAL6 & 3754 & 3754 & 19,477 & 3380 \\ POTENTIAL10 & 3502 & 3502 & 13,129 & 2538 \\ HANSEN & 3407 & 3407 & 8345 & 2795 \\ HARTMAN3 & 3251 & 3251 & 7587 & 2911 \\ HARTMAN6 & 3639 & 3639 & 13,342 & 3408 \\ RASTRING & 2878 & 4499 & 6463 & 2806 \\ ROSENBROCK8 & 5859 & 22,384 & 24,123 & 5859 \\ ROSENBROCK16 & 8528 & 23,513 & 24,132 & 8528 \\ SHEKEL5 & 3381 & 3381 & 16,781 & 2778 \\ SHEKEL7 & 3600 & 3600 & 17,989 & 3324 \\ SHEKEL10 & 3258 & 3258 & 19,227 & 2866 \\ SINU4 & 2923 & 2923 & 9767 & 2231 \\ SINU8 & 2934 & 2934 & 10,714 & 2502 \\ TEST2N4 & 3623 & 3623 & 9619 & 3603 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Experiments using the variety of termination rules and the GWO optimization method. Figure 4: Statistical comparison of function calls with different termination rule of multistart. The proposed termination rule was also applied to the current parallel optimization technique for the test functions described previously. The experimental results for the so-called _DoubleBox_ stopping rule and the given parallel method are shown in Table 8. The columns in the table stand for the following:

1. Column Problem denotes the test function used. 2. Column \(1\times 500\) denotes the application of the proposed technique with one processing unit and 500 particles. 3. Column \(2\times 250\) stands for the usage of two processing units.