## 3 Methodology
\begin{table}
\begin{tabular}{c l l l} \hline \hline
**Denotation** & **Test Function** & \(\mathbf{\upbeta}\) & **Modality** \\ \hline F1 & Shifted Sphere Function & \([-100,100]~{}^{D}\) & Unimodal \\ F2 & Shifted Schwefel’s Problem 1.2 & \([-100,100]~{}^{D}\) & Unimodal \\ F3 & Shifted Rotated Hight Conditioned & \([-100,100]~{}^{D}\) & Unimodal \\ F4 & Shifted Schwefel’s Problem 1.2 with & \([-100,100]~{}^{D}\) & Unimodal \\ F5 & Schwefel’s Problem 2.6 with Global & \([-100,100]~{}^{D}\) & Unimodal \\ F6 & Shifted [PERSON]’s Function & \([-100,100]~{}^{D}\) & Basic Multimodal \\ F7 & Shifted Rotated Griewank’s Function & \([0,600]~{}^{D}\) & Basic Multimodal \\ F8 & Shifted Rotated Ackley’s Function with & \([-32,32]~{}^{D}\) & Basic Multimodal \\ F9 & Shifted Rastrigin’s Function & \([-5,5]~{}^{D}\) & Basic Multimodal \\ F10 & Shifted Rotated Rastrigin’s Function & \([-5,5]~{}^{D}\) & Basic Multimodal \\ F11 & Shifted Rotated Weierstrass Function & \([-0.5,0.5]~{}^{D}\) & Basic Multimodal \\ F12 & [PERSON]’s Problem 2.13 & \([-\Pi,\Pi]~{}^{D}\) & Basic Multimodal \\ F13 & Shifted Expanded Griewank’s plus & \([-5,5]~{}^{D}\) & Expanded Multimodal \\ F14 & Shifted Rotated Expanded Scaffer’s \(f6\) & \([-100,100]~{}^{D}\) & Expanded Multimodal \\ F15 & Hybrid Composition Function & \([-5,5]~{}^{D}\) & Hybrid Composition \\ F16 & Rotated Version of Hybrid & \([-5,5]~{}^{D}\) & Hybrid Composition \\ F17 & Composition Function \(f15\) with Noise & \([-5,5]~{}^{D}\) & Hybrid Composition \\ F18 & Rotated Hybrid Composition Function & \([-5,5]~{}^{D}\) & Hybrid Composition \\ F19 & Rotated Hybrid Composition Function & \([-5,5]~{}^{D}\) & Hybrid Composition \\ F20 & Rotated Hybrid Composition Function & \([-5,5]~{}^{D}\) & Hybrid Composition \\ F21 & Rotated Hybrid Composition Function & \([-5,5]~{}^{D}\) & Hybrid Composition \\ F22 & Rotated Hybrid Composition Function & \([-5,5]~{}^{D}\) & Hybrid Composition \\ F23 & Non-continuous Rotated Hybrid & \([-5,5]~{}^{D}\) & Hybrid Composition \\ Composition Function & & \([-5,5]~{}^{D}\) & Hybrid Composition \\ F24 & Rotated Hybrid Composition Function & \([-5,5]~{}^{D}\) & Hybrid Composition \\ F25 & Rotated Hybrid Composition Function & Initialize population in & Hybrid Composition \\  & without Bounds & \([2,5]~{}^{D}\) & but no exact search \\  & & range set \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Details on the benchmark functions. ## 5 Results and Analysis

The difference between current fitness value and optimum value, also known as error value, is used to compare the algorithms' performance. The average errors of the independent runs of all algorithms for \(10D\), \(30D\) and \(50D\) problems are summarized in Tables 4-6. The last rows in Tables 4-6 show the frequency of having the best result for the algorithms, \(h_{win}\). Based on the results in Tables 4-6, DE cannot find the solutions for functions F15-F17 because its solutions have undefined numeric results for the functions that are divided by zero. When the settings are \(10D\) and \(N\) is \(25\) (\(10D25N\)), SaDE, DE-PSO and FIPSaDE have the highest \(h_{win}\), which is \(6\). This means the three algorithms have compatible performances in solving the problems. An interesting finding from the comparison of \(h_{win}\) for SaDE, DE-PSO and FIPSaDE for the setting of \(10D25N\) is that they have the lowest error for different functions. This finding may indicate that they perform well for different characteristics of problems. As the combinations of \(D\) and \(N\) increase to \(30D30N\) and \(50D50N\), FIPSaDE shows distinctive performances compared to the other algorithms by having the highest \(h_{win}\) in both settings. FIPSaDE has \(h_{win}=9\) and \(h_{win}=12\) in \(30D30N\) and \(50D50N\), respectively. In other words, FIPSaDE's performances improves with the increase of problem dimensionality and population size. In contrast, the performances of FIPS are the worst among the algorithms for the three settings.