# Combining Parallel Stochastic Methods and Mixed Termination Rules in Optimization
## 2 Adopted Algorithms
### The Method Multistart
During parallel execution, the total sum of calls is computed across all computing units nodes, ensuring efficient load distribution and resource utilization. However, the algorithm terminates at the computing unit that identifies the optimal solution, signaling the achievement of the desired convergence criterion. In Table 3, we observe the performance of all termination rules for the DE method. From these, both statistical and quantitative comparisons arise, depicted in Figure 2. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Name** & **Formula** & **Dimension** \\ \hline Gkls [84] & \(f(x)=\mathrm{Gkls}(x,n,w)\) & \(n=2,3\ w=50,100\) \\ \hline [PERSON] & \(f(x)=\sum_{i=1}^{5}i\cos[(i-1)x_{1}+i]\sum_{j=1}^{5}j\cos[(j+1)x_{2}+j]\) & 2 \\ \hline [PERSON] & \(f(x)=-\sum_{i=1}^{4}c_{i}\exp\left(-\sum_{j=1}^{3}a_{ij}\big{(}x_{j}-p_{ij} \big{)}^{2}\right)\) & 3 \\ \hline [PERSON] & \(f(x)=-\sum_{i=1}^{4}c_{i}\exp\left(-\sum_{j=1}^{6}a_{ij}\big{(}x_{j}-p_{ij} \big{)}^{2}\right)\) & 6 \\ \hline [PERSON] & \(f(x)=\sum_{i=1}^{n}(10^{6})^{\frac{i-1}{2}}x_{i}^{2}\) & 10 \\ \hline [PERSON] [85] & \(V_{LJ}(r)=4e\left[\left(\frac{r}{r}\right)^{12}-\left(\frac{r}{r}\right)^{6}\right]\) & \(n=9,15,21,30\) \\ \hline [PERSON] & \(f(x)=x_{1}^{2}+x_{2}^{2}-\cos(18x_{1})-\cos(18x_{2})\) & 2 \\ \[PERSON] & \(f(x)=\sum_{i=1}^{n-1}\left(100\big{(}x_{i+1}-x_{i}^{2}\big{)}^{2}+(x_{i}-1)^{2 }\right)\), & \(-30\leq x_{i}\leq 30\) & \(n=4,8,16\) \\ \[PERSON] & \(f(x)=-\sum_{i=1}^{5}\frac{1}{(x-a_{i})(-a_{i})^{i}+c_{i}}\) & 4 \\ \[PERSON] & \(f(x)=-\sum_{i=1}^{7}\frac{1}{(x-a_{i})(x-a_{i})^{i}+c_{i}}\) & 4 \\ \hline Shekel10 & \(f(x)=-\sum_{i=1}^{10}\frac{1}{(x-a_{i})(x-a_{i})^{i}+c_{i}}\) & 4 \\ \hline Sinusoidal [86] & \(f(x)=-(2.5\prod_{i=1}^{n}\sin(x_{i}-2)+\prod_{i=1}^{n}\sin(5(x_{i}-2)))\), & \(0\leq x_{i}\leq\pi\) & \(n=4,8\) \\ \hline Test2N & \(f(x)=\frac{1}{2}\sum_{i=1}^{n}x_{4}^{4}-16x_{2}^{2}+5x_{i}\) & \(n=4,9\) \\ \hline Test30N & \(\frac{1}{10}\sin^{2}(3\pi x_{1})\sum_{i=2}^{n-1}\left((x_{i}-1)^{2}\big{(}1+ \sin^{2}(3\pi x_{i+1})\big{)}\right)+(x_{n}-1)^{2}\big{(}1+\sin^{2}(2\pi x_{n} )\big{)}\) & \(n=3,4\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: _Cont._ \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Problem** & **Best** & **Mean** & **DoubleBox** & **All** \\ \hline BF1 & 6761 & 10,511 & 10,369 & 6114 \\ BF2 & 7693 & 9222 & 11,524 & 7560 \\ BRANIN & 4639 & 10,982 & 4907 & 4289 \\ CAMEL & 6116 & 13,290 & 9905 & 5940 \\ CIGAR10 & 7647 & 4348 & 21,116 & 3111 \\ CM4 & 7913 & 4175 & 11,851 & 4079 \\ DISCUS10 & 5191 & 2042 & 13,614 & 2034 \\ EASOM & 1917 & 15,103 & 1721 & 1721 \\ ELP10 & 6046 & 24,675 & 12,519 & 6031 \\ EXP4 & 5216 & 18,151 & 6385 & 5040 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Average function calls for the Differential evolution method, using different termination rules. As it is evident, the combination of the stopping rules reduces the required number of function calls that are required to obtain the global minimum by the DE method. The proposed termination scheme outperforms by a percentage that varies from 13% to 73% the other termination rules used here. In addition, in order to ascertain which of the three termination rules has the greatest effect on the termination with the mixed rule, the rule responsible for the termination of the DE optimization technique was recorded for all functions of the test set. The results from this recording are shown in Table 4. The