# Combining Parallel Stochastic Methods and Mixed Termination Rules in Optimization
## 2 Adopted Algorithms
### The Method Multistart
In each unit, the number of particles was set to 250. 4. Column \(5\times 100\) denotes the incorporation of 5 processing units into the proposed method. In each processing unit the number of particles was set to 100. 5. Column \(10\times 50\) stands for the usage of 10 processing units. In each processing unit, the number of particles was set to 50. In this experiment, the total number of particles was set to 500, in order to have credibility in the values recorded. The results indicate that the increase in processing units may reduce the required number of function calls. The same series of experiments was also conducted using the so-called _mean fitness_ termination rule, which has been described in Equations (2) and (3). The obtained results are presented in Tables 9 and 10. And, in this series of experiments, the columns of the table retain the same meaning as in Table 8. Furthermore, once again, it is observed that the increase in the number of computing units significantly reduces the required number of function calls to find the global minimum. Additionally, there is a significant reduction in the number of function calls required to be compared to the previous termination rule. Finally, the proposed termination rule is utilized in the parallel optimization technique and the experimental results are outlined in Table 11. \begin{table}
\begin{tabular}{l c c c c} \hline
**Problem** & **Best** & **Mean** & **DoubleBox** & **All** \\ \hline TEST2N9 & 4327 & 4327 & 21,381 & 4207 \\ TEST30N3 & 3225 & 3237 & 3513 & 2769 \\ TEST30N4 & 3324 & 3324 & 6124 & 3144 \\
**Total** & **152,006** & **236,870** & **451,937** & **144,460** \\ \hline \end{tabular}
\end{table} 
Table 7: _Cont._

Figure 5: Statistical comparison for the GWO optimizer using different termination rules. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Problem** & **1 \(\times\) 500** & **2 \(\times\) 250** & **5 \(\times\) 100** & **10 \(\times\) 50** \\ \hline BF1 & 36,018 & 28,282 & 18,419 & 10,859 \\ BF2 & 42,191 & 37,555 & 16,568 & 10,710 \\ BRANIN & 45,165 & 45,726 & 45,757 & 34,381 \\ CAMEL & 54,786 & 54,782 & 54,253 & 53,839 \\ CIGAR10 & 18,387 & 18,300 & 18,307 & 11,950 \\ CM4 & 60,796 & 60,501 & 60,287 & 60,200 \\ DISCUS10 & 25,115 & 24,586 & 14,565 & 8957 \\ EASOM & 40,376 & 40,211 & 40,100 & 40,099 \\ ELP10 & 20,193 & 14,745 & 11,211 & 7691 \\ EXP4 & 54,779 & 54,676 & 54,416 & 54,412 \\ EXP16 & 52,768 & 52,205 & 52,563 & 52,560 \\ GKLS250 & 51,980 & 51,240 & 51,233 & 51,231 \\ GKLS350 & 48,577 & 48,761 & 48,702 & 4100 \\ GRIEWANK2 & 56,767 & 56,690 & 51,170 & 26,076 \\ GRIEWANK10 & 20,941 & 20,721 & 14,870 & 11,765 \\ POTENTIAL3 & 49,418 & 49,800 & 49,109 & 49,002 \\ POTENTIAL5 & 59,827 & 59,656 & 59,367 & 59,322 \\ POTENTIAL6 & 62,720 & 62,398 & 62,294 & 62,280 \\ POTENTIAL10 & 72,899 & 72,745 & 72,943 & 72,578 \\ HANSEN & 45,064 & 45,689 & 45,510 & 45,424 \\ HARTMAN3 & 47,307 & 47,112 & 47,111 & 47,100 \\ HARTMAN6 & 49,370 & 49,302 & 49,312 & 49,222 \\ RASTRIGIN & 57,617 & 56,577 & 56,513 & 56,400 \\ ROSENBROCK8 & 22,997 & 22,903 & 14,139 & 14,001 \\ ROSENBROCK16 & 42,117 & 34,856 & 21,392 & 13,722 \\ SHEKEL5 & 50,631 & 50,198 & 50,233 & 44,662 \\ SHEKEL7 & 51,212 & 51,200 & 51,200 & 45,610 \\ SHEKEL10 & 51,741 & 51,607 & 51,587 & 47,761 \\ SINU4 & 48,140 & 48,100 & 48,111 & 48,099 \\ SINU8 & 48,761 & 48,434 & 48,423 & 48,204 \\ TEST2N4 & 48,904 & 48,307 & 48,354 & 48,354 \\ TEST2N9 & 48,838 & 48,555 & 48,600 & 48,335 \\ TEST30N3 & 51,640 & 51,307 & 51,194 & 51,143 \\ TEST30N4 & 48,992 & 48,549 & 48,540 & 48,487 \\
**Total** & **1,587,034** & **1,556,276** & **1,476,353** & **1,338,536** \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Experiments using the proposed optimization technique and the DoubleBox stopping rule. The experiment was performed on the test functions described previously. Numbers in cells denote average function calls.