# A Novel Complex-Valued Encoding Grey Wolf Optimization Algorithm
## 3 Complex-Valued Encoding Grey Wolf Optimization Algorithm (CGWO)
### CGWO Algorithm
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline
**Dim = 10** & \(\mathbf{F_{1}}\) & \(\mathbf{F_{2}}\) & \(\mathbf{F_{3}}\) & \(\mathbf{F_{4}}\) & \(\mathbf{F_{5}}\) & \(\mathbf{F_{6}}\) & \(\mathbf{F_{7}}\) & \(\mathbf{F_{8}}\) \\ \hline CGWO \(\Rightarrow\). DGWO & 2.4426E\(-\)08 & 8.0065E\(-\)09 & 6.2771E\(-\)08 & 6.7956E\(-\)08 & 6.7956E\(-\)08 & 6.7956E\(-\)08 & 1.2346E\(-\)07 & 1.6571E\(-\)07 \\ CGWO \(\Rightarrow\). GWO & 2.4426E\(-\)08 & 8.0065E\(-\)09 & 6.2771E\(-\)08 & 6.7956E\(-\)08 & 2.1393E\(-\)03 & 3.5070E\(-\)01 & 4.1658E\(-\)05 & 1.1045E\(-\)05 \\ CGWO \(\Rightarrow\). GGSA & 2.4426E\(-\)08 & 8.0065E\(-\)09 & 6.2771E\(-\)08 & 6.7956E\(-\)08 & 7.9479E\(-\)07 & 8.0065E\(-\)09 & 6.7956E\(-\)08 & 6.7956E\(-\)08 \\ CGWO \(\Rightarrow\). ABC & 2.4426E\(-\)08 & 8.0065E\(-\)09 & 6.2771E\(-\)08 & 6.7956E\(-\)08 & 6.7956E\(-\)08 & 9.1728E\(-\)08 & 6.7956E\(-\)08 & 9.1728E\(-\)08 \\ \hline
**Dim = 10** & \(\mathbf{F_{3}}\) & \(\mathbf{F_{10}}\) & \(\mathbf{F_{11}}\) & \(\mathbf{F_{2}}\) & \(\mathbf{F_{3}}\) & \(\mathbf{F_{4}}\) & \(\mathbf{F_{5}}\) & \(\mathbf{F_{6}}\) \\ \hline CGWO \(\Rightarrow\). DGWO & 8.0065E\(-\)09 & 2.4037E\(-\)08 & 1.5149E\(-\)08 & 6.7956E\(-\)08 & 8.0065E\(-\)09 & 2.5629E\(-\)07 & 6.3761E\(-\)08 & 1.0646E\(-\)07 \\ CGWO \(\Rightarrow\). GWO & 1.6259E\(-\)01 & 1.7005E\(-\)07 & 8.2529E\(-\)04 & 6.9166E\(-\)07 & 8.0065E\(-\)09 & 7.5774E\(-\)06 & 1.1613E\(-\)07 & 4.9864E\(-\)02 \\ CGWO \(\Rightarrow\). GGSA & 7.3831E\(-\)09 & 2.4037E\(-\)08 & 1.5149E\(-\)08 & 6.7956E\(-\)08 & 8.0065E\(-\)09 & 3.7499E\(-\)04 & 1.5605E\(-\)07 & 3.2348E\(-\)01 \\ CGWO \(\Rightarrow\). ABC & 8.0065E\(-\)09 & 2.4037E\(-\)08 & 1.5149E\(-\)08 & 6.7956E\(-\)08 & 8.0065E\(-\)09 & 1.9177E\(-\)07 & 7.4151E\(-\)08 & 7.1135E\(-\)03 \\ \hline \end{tabular}
\end{table} 
Table 6: p-values produced by Wilcoxon rank-sum test comparing CGWO (complex-valued encoding grey wolf optimization) \(\Rightarrow\). DGWO (differential evolution GWO), GWO, GGSA(gbest-guided gravitational search algorithm), and ABC (artificial bee colony optimization)minimization results of the multimodal benchmark functions at Dim = 10 (\(p\geq 0.05\)). \begin{table}
\begin{tabular}{c c c c c c c c c} \hline
**Dim = 30** & \(\mathbf{F_{1}}\) & \(\mathbf{F_{2}}\) & \(\mathbf{F_{3}}\) & \(\mathbf{F_{4}}\) & \(\mathbf{F_{5}}\) & \(\mathbf{F_{6}}\) & \(\mathbf{F_{7}}\) & \(\mathbf{F_{8}}\) \\ \hline CGWO \(\Rightarrow\). DGWO & 6.7956E\(-\)08 & 6.7956E\(-\)08 & 6.7956E\(-\)08 & 6.7956E\(-\)08 & 6.7956E\(-\)08 & 2.0616E\(-\)06 & 7.8908E\(-\)08 \\ CGWO \(\Rightarrow\). GWO & 6.7956E\(-\)08 & 6.7956E\(-\)08 & 6.7956E\(-\)08 & 6.7956E\(-\)08 & 5.1153E\(-\)03 & 6.7956E\(-\)08 & 2.5629E\(-\)07 & 2.3025E\(-\)05 \\ CGWO \(\Rightarrow\). GGSA & 6.7956E\(-\)08 & 6.7956E\(-\)08 & 6.7956E\(-\)08 & 6.7956E\(-\)08 & 6.7956E\(-\)08 & 6.7956E\(-\)08 & 6.7956E\(-\)08 & 6.7956E\(-\)08 \\ CGWO \(\Rightarrow\). ABC & 6.7956E\(-\)08 & 6.7956E\(-\)08 & 6.7956E\(-\)08 & 6.7956E\(-\)08 & 6.7956E\(-\)08 & 6.7956E\(-\)08 & 6.7956E\(-\)08 \\ \hline
**Dim = 30** & \(\mathbf{F_{5}}\) & \(\mathbf{F_{10}}\) & \(\mathbf{F_{11}}\) & \(\mathbf{F_{2}}\) & \(\mathbf{F_{3}}\) & \(\mathbf{F_{4}}\) & \(\mathbf{F_{5}}\) & \(\mathbf{F_{6}}\) \\ \hline CGWO \(\Rightarrow\). DGWO & 1.5149E\(-\)08 & 4.7371E\(-\)08 & 1.1267E\(-\)08 & 6.7956E\(-\)08 & 6.7956E\(-\)08 & 6.7956E\(-\)08 & 6.7956E\(-\)08 & 6.7956E\(-\)08 \\ CGWO \(\Rightarrow\). GWO & 3.6853E\(-\)05 & 3.8179E\(-\)08 & 1.8196E\(-\)02 & 6.7956E\(-\)08 & 6.7956E\(-\)08 & 6.7956E\(-\)08 & 6.7956E\(-\)08 & 4.7348E\(-\)01 \\ CGWO \(\Rightarrow\). GGSA & 1.4970E\(-\)08 & 4.7371E\(-\)08 & 1.1267E\(-\)08 & 1.8074E\(-\)05 & 6.7956E\(-\)08 & 6.7956E\(-\)08 & 6.7956E\(-\)08 & 6.7956E\(-\)08 \\ CGWO \(\Rightarrow\). ABC & 1.5149E\(-\)08 & 4.7371E\(-\)08 & 1.1267E\(-\)08 & 6.7956E\(-\)08 & 6.7956E\(-\)08 & 6.7956E\(-\)08 & 6.7956E\(-\)08 \\ \hline \end{tabular}
\end{table} 
Table 7: p-values produced by Wilcoxon rank-sum test comparing CGWO \(\Rightarrow\). DGWO, GWO, GGSA, and ABC at Dim = 30 (\(p\geq 0.05\) have been underlined).