# 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} \hline \hline
**Dim = 400** & \(\mathbf{F_{1}}\) & \(\mathbf{F_{2}}\) & \(\mathbf{F_{3}}\) & \(\mathbf{F_{4}}\) & \(\mathbf{F_{5}}\) & \(\mathbf{F_{6}}\) & \(\mathbf{F_{7}}\) & \(\mathbf{F_{8}}\) \\ \hline CGWO vs. DGWO & 6.7956E\(-\)08 & 6.7956E\(-\)08 & 6.7956E\(-\)08 & 6.7956E\(-\)08 & 6.7956E\(-\)08 & 6.7956E\(-\)08 & 6.7956E\(-\)08 & 9.1266E\(-\)07 \\ CGWO vs. GWO & 6.7956E\(-\)08 & 6.7956E\(-\)08 & 6.7956E\(-\)08 & 1.2346E\(-\)07 & 7.3527E\(-\)01 & 6.7956E\(-\)08 & 9.1728E\(-\)08 & 7.2045E\(-\)02 \\ CGWO vs. GGSA & 6.7956E\(-\)08 & 6.7956E\(-\)08 & 7.3527E\(-\)01 & 6.7956E\(-\)08 & 6.7956E\(-\)08 & 6.7956E\(-\)08 & 1.9177E\(-\)07 & 2.3025E\(-\)05 \\ CGWO vs. ABC & 6.7956E\(-\)08 & 6.7956E\(-\)08 & 6.7956E\(-\)08 & 6.7956E\(-\)08 & 6.7956E\(-\)08 & 6.7956E\(-\)08 & 3.9388E\(-\)07 \\ \hline
**Dim = 400** & \(\mathbf{F_{3}}\) & \(\mathbf{F_{10}}\) & \(\mathbf{F_{11}}\) & \(\mathbf{F_{12}}\) & \(\mathbf{F_{13}}\) & \(\mathbf{F_{14}}\) & \(\mathbf{F_{15}}\) & \(\mathbf{F_{16}}\) \\ \hline CGWO vs. DGWO & 1.4309E\(-\)07 & 6.7956E\(-\)08 & 3.3727E\(-\)08 & 6.7956E\(-\)08 & 6.7956E\(-\)08 & 6.7956E\(-\)08 & 6.7956E\(-\)08 & 6.7956E\(-\)08 \\ CGWO vs. GWO & 7.1135E\(-\)03 & 6.7956E\(-\)08 & 4.4078E\(-\)07 & 6.7956E\(-\)08 & 6.7956E\(-\)08 & 6.9166E\(-\)07 & 6.7956E\(-\)08 & 6.7956E\(-\)08 \\ CGWO vs. GGSA & 1.6098E\(-\)04 & 6.7956E\(-\)08 & 3.3727E\(-\)08 & 6.7956E\(-\)08 & 6.7956E\(-\)08 & 7.5788E\(-\)04 & 6.7956E\(-\)08 & 6.7956E\(-\)08 \\ CGWO vs. ABC & 6.7956E\(-\)08 & 6.7956E\(-\)08 & 3.3727E\(-\)08 & 6.7956E\(-\)08 & 6.7956E\(-\)08 & 6.7956E\(-\)08 & 6.7956E\(-\)08 \\ \hline \hline \end{tabular}
\end{table} 
Table 12: \(p\)-values produced by Wilcoxon rank-sum test comparing CGWO vs. DGWO, GWO, GGSA, and ABC at Dim = 400 (\(p\geq 0.05\) have been underlined). ### Unimodal Benchmark Functions

The unimodal benchmarks have only one global solution without any local optima for them. Consequently, they are useful in examining the convergence capability of heuristic optimization. The results of unimodal benchmark functions are shown in Table 3. As shown in this table, for 30-D unimodal benchmark functions, CGWO has the best results for all benchmark functions. For 10-D, 50-D, 100-D, 200-D, 300-D, and 400-D, this table shows that the CGWO algorithm outperforms the other algorithms on the majority of the unimodal benchmark test functions. Therefore, the proposed algorithm has high performance in finding the global solution of unimodal benchmark test functions. According to the \(p\) values of \(F_{1}\sim F_{8}\) with different dimensions in Tables 5-11 CGWO achieves significant improvement across the majority of dimensions compared to the other algorithms. Hence, this proves that CGWO has better performance than the other algorithms in foring for a global optimum solution of unimodal benchmark functions. Figure 1 shows the averaged convergence curves of unimodal benchmark functions with 30-D over 20 independent runs. As shown from these curves, the CGWO has the fastest convergence rate. Figure 2 indicates the anova tests of the global minimum on unimodal functions with 30-D, which shows that CGWO is the most robust method. For \(F_{1}\sim F_{8}\) functions, the other algorithms that we compared (namely, DGWO, GWO, GGSA, ABC) were only robust for a few functions and were not able to be robust for all functions. On the basis of Tables 3 and 5-11 and Figures 1 and 3 it can be concluded that the CGWO tends to find the best solution in unimodal functions with a very fast convergence rate and a more stable performance. Figure 1: Convergence behavior of ABC, GGSA, GWO, DGWO, and CGWO on unimodal functions with Dim = 30. Figure 2: _Cont_

### Multimodal Benchmark Functions

The results of the multimodal benchmark functions are provided in Table 4. Note that multimodal functions have many local minima with the number increasing exponentially with dimension. Therefore, they are very suitable to test the exploration capability of algorithms to avoid local minima.