# Combining Parallel Stochastic Methods and Mixed Termination Rules in Optimization
## 2 Adopted Algorithms
### The Method Multistart
### Test Functions

The test functions [81; 82] presented below exhibit varying levels of difficulty in solving them; hence, a periodic local optimization mechanism has been incorporated. Periodic local optimization plays a crucial role in increasing the success rate in locating the minimum of functions. This addition appears to lead to a success rate approaching 100% for all functions, regardless of their characteristics such as dimensionality, minima, scalability, and symmetry. A study by [PERSON] and colleagues [83] specifically examines the issue of symmetry and asymmetry in the test functions. The test functions used in the conducted experiments are shown in \[\delta^{(k)}=\left|\sum_{i=1}^{\text{NP}}\left|f_{i}^{(k)}\right|-\sum_{i=1}^{ \text{NP}}\left|f_{i}^{(k-1)}\right|\right| \tag{3}\]0 Generally, the region of attraction \(A(x^{*})\) of a local minimum \(x^{*}\) is defined as:

\[\delta^{(k)}=\left|\sum_{i=1}^{\text{NP}}\left|f_{i}^{(k)}\right|-\sum_{i=1}^{ \text{NP}}\left|f_{i}^{(k-1)}\right|\right| \tag{3}\]1

where \(L(x)\) is a local search procedure, such as BFGS. Global optimization methods usually find points which are in the region of attraction \(A(x^{*})\) of a local minimum \(x^{*}\), but not necessarily the local minimum itself. For this reason, at the end of their execution, a local minimization method is applied in order to ensure the exact finding of a local minimum. Of course, this does not imply that the minimum that will be located will be the total minimum of the function, since a function can contain tens or even hundreds of local minima. \begin{table}
\begin{tabular}{c c c} \hline
**Parameter** & **Value** & **Explanation** \\ \hline \(N_{p}\) & 120 & Total particles for PSO \\ \(N_{d}\) & 120 & Total elements for DE \\ \(N_{m}\) & 120 & Total elements for Multistart \\ \(N_{k}\) & 200 & Maximum number of iterations \\ \(N_{s}\) & 15 & Similarity max count \\ \(F\) & 0.8 & Differential weight for DE \\ \(CR\) & 0.9 & Crossover Probability for DE \\ \(P_{m}\) & 0.005 & Local search rate for DE and PSO \\ \(C_{1},C_{2}\) & 0.5 & Parameters of PSO \\ \hline \end{tabular}
\end{table} 
Table 1: The following parameters were considered for conducting the experiments. \begin{table}
\begin{tabular}{c c c} \hline
**Name** & **Formula** & **Dimension** \\ \hline Cigar10 & \(f(x)=x_{1}^{2}+10^{6}\sum_{i=2}^{n}x_{i}^{2}\) & 10 \\ \hline BF1 & \(f(x)=x_{1}^{2}+2x_{2}^{2}-\frac{3}{10}\cos(3\pi x_{1})-\frac{4}{10}\cos(4\pi x _{2})+\frac{7}{10}\) & 2 \\ \hline BF2 & \(f(x)=x_{1}^{2}+2x_{2}^{2}-\frac{3}{10}\cos(3\pi x_{1})\cos(4\pi x_{2})+\frac{3 }{10}\) & 2 \\ \hline Branin & \(f(x)=\left(x_{2}-\frac{5.1}{4\pi^{2}}x_{1}^{2}+\frac{5}{\pi}x_{1}-6\right)^{2} +10\big{(}1-\frac{1}{8\pi}\big{)}\cos(x_{1})+10\) & 2 \\ \hline CM & \(f(x)=\sum_{i=1}^{n}x_{i}^{2}-\frac{1}{10}\sum_{i=1}^{n}\cos(5\pi x_{i})\) & 4 \\ \hline Discus & \(f(x)=10^{6}x_{1}^{2}+\sum_{i=2}^{n}x_{i}^{2}\) & 10 \\ \hline Easom & \(f(x)=-\cos(x_{1})\cos(x_{2})\exp\big{(}(x_{2}-\pi)^{2}-(x_{1}-\pi)^{2}\big{)}\) & 2 \\ \hline Exp & \(f(x)=-\exp\big{(}-0.5\sum_{i=1}^{n}x_{i}^{2}\big{)},\quad-1\leq x_{i}\leq 1\) & \(n=4,16\) \\ \hline Griewank2 & \(f(x)=1+\frac{1}{200}\sum_{i=1}^{2}x_{i}^{2}-\prod_{i=1}^{2}\frac{\cos(x_{i})}{ \sqrt{(i)}}\) & 2 \\ \hline Griewank10 & \(\text{f}(x)=1+\frac{1}{200}\sum_{i=1}^{10}x_{i}^{2}-\prod_{i=1}^{10}\frac{\cos(x_{ i})}{\sqrt{(i)}}\) & 10 \\ \hline \end{tabular}
\end{table} 
Table 2: The test functions used in the conducted experiments. ### Experimental Results

All tables presented here demonstrate the average function calls obtained for each test problem. The average function can be considered as a measure of the speed of each method, since each function has a different complexity and therefore, the average execution time would not be a good indication of the computational time required. In the first set of experiments we examined if the presence of the suggested stopping rule, which is the combination of a series of stopping rules, affects the optimization techniques without parallelization.