## 5 Experimental Results and Analysis
### Comparison with DE, PSO, and Three State-of-the-Art DE Variants  
In order to evaluate the performance of the HE-DEPSO algorithm, its optimization results and convergence properties are compared and analyzed with respect to those obtained by the algorithms DE, PSO, CoDE, SHADE, and DEPSO, on the set of CEC 2017 test functions. Tables 4 and 5 report the statistical results for each algorithm at \(D=10\) and \(D=30\), respectively. The error measure of the solution \(|F_{min}-F(X^{*})|\), where \(F_{min}\) is the known solution of the problem and \(X^{*}\) is the best solution found in each iteration of each algorithm, was used to obtain these results. These tables show the mean and standard deviation (Std) of the solution error measure for each function and algorithm over the 31 independent runs and the ranking achieved based on the mean value obtained. The best results are highlighted in bold. A non-parametric Wilcoxon rank sum test, with a confidentiality level of \(\alpha=0.05\), was performed to identify statistically significant differences between the results obtained by the HE-DEPSO algorithm and the results obtained by the other algorithms. In these tables, the statistical significance related to the performance of the HE-DEPSO algorithm is represented by the symbols \"+\", \"\(\approx\)\", and \"\(-\)\", which indicate that the performance of the HE-DEPSO algorithm is \"better/similar/worse\" than the algorithm to which it is compared. The row \"W/T/L\" counts the total number of \"+\", \"\(\approx\)\", and \"\(-\)\", respectively. \begin{table}
\begin{tabular}{l l l l l l l l} \hline \hline
**Function** & **Metrics** & **HE-DEPSO** & **DEPSO** & **SHADE** & **CoDE** & **DE** & **PSO** \\ \hline \multirow{3}{*}{\(F_{1}\)} & Mean & **0.0000** & 1.8337 \(\times 10^{-15}\) & **0.0000** & 1.1580 \(\times 10^{-10}\) & 3.5103 \(\times 10^{-5}\) & 1.1366 \(\times 10^{10}\) \\  & Std & **0.0000** & 4.8427 \(\times 10^{-15}\) & **0.0000** & 6.0507 \(\times 10^{-11}\) & 2.0793 \(\times 10^{-5}\) & 9.7631 \(\times 10^{9}\) \\  & Rank & **1** & 4 & **1** & 3 & 2 & 5 \\  & Significance & & \(\approx\) & \(\approx\) & \(+\) & \(+\) & \(+\) \\ \hline \multirow{3}{*}{\(F_{1}\)} & Mean & **9.1683 \(\times 10^{-16}\)** & 1.3752 \(\times 10^{-14}\) & 1.6503 \(\times 10^{-14}\) & 2.0001 \(\times 10^{-9}\) & 1.7359 \(\times 10^{-4}\) & 1.5886 \(\times 10^{10}\) \\ \cline{1-1} \cline{2-7}  & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \\ \end{tabular}
\end{table} 
Table 4: Comparison of results between HE-DEPSO, DE, PSO, and three advanced variants of the DE algorithm on the set of CEC 2017 test functions with \(D=10\). \begin{table}
\begin{tabular}{c c} \hline \hline
**Algorithm** & **Parameters** \\ \hline HE-DEPSO & \(p_{min}=0.1\), \(p_{max}=0.4\), \(\tau=1.8\), \(c1=c2=2\), \(\omega\in[0.4,0.9]\), \\  & \(H=NP\), \(M_{F}(1:H)=0.5\), \(M_{Cr}(1:H)=0.8\) \\ \hline DEPSO & \(c1=c2=2\), \(\omega\in[0.4,0.9]\), \(Cr[0.3,1.0]\), \(F[0.1,0.8]\), \\  & \(NS_{max}=5\), \(\gamma=0.001\), \(\tau=1.8\), \(SEP=0.3\cdot NP\) \\ \hline SHADE & \(H=NP\), \(M_{F}(1:H)=M_{Cr}(1:H)=0.5\) \\ \hline CoDE & (1) [\(F=1.0,Cr=0.1]\), (2) [\(F=1.0,Cr=0.9]\), (3) [\(F=0.8,Cr=0.2]\) \\ \hline DE & \(Cr=0.9\), \(F=0.5\) \\ \hline PSO & \(\omega\in[0.4,0.9]\), \(c1=c2=2\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameter settings of all algorithms.