# A Novel Complex-Valued Encoding Grey Wolf Optimization Algorithm
## 3 Complex-Valued Encoding Grey Wolf Optimization Algorithm (CGWO)
### CGWO Algorithm
In order to determine whether the results of CGWO differ from the best results of the other algorithms in a statistical method, a nonparametric test, which is known as [PERSON]'s rank-sum test [28; 29; 30; 31], is performed at 5% significance level. The calculated \(p\) values in [PERSON]'s rank-sum test comparing CGWO and the other algorithms over all the benchmark functions are given in Tables 6-12. Usually, \(p\) values < 0.05 can be considered as sufficient evidence against the null hypothesis. \begin{table}
\begin{tabular}{c c c c} \hline
**Benchmark Functions** & **Range** & **Dim** & \(f_{\text{min}}\) \\ \hline \multicolumn{4}{c}{_Unimodal functions_} & 0 \\ \(F_{1}\): Sphere function & [\(-\)100,100] & 10,30,50,100,200,300,400 & 0 \\ \(F_{2}\): [PERSON]’s problem 2.22 & [\(-\)10,10] & 10,30,50,100,200,300,400 & 0 \\ \(F_{3}\): [PERSON]’s problem 2.21 & [\(-\)10,100] & 10,30,50,100,200,300,400 & 0 \\ \(F_{4}\): Generalized [PERSON]’s function & [\(-\)30,30] & 10,30,50,100,200,300,400 & 0 \\ \(F_{5}\): Quartet function _ie_, Nose & [\(-\)128,128] & 10,30,50,100,200,300,400 & 0 \\ \(F_{6}\): Shifted sphere function & [\(-\)100,100] & 10,30,50,100,200,300,400 & \(-\)450 \\ \(F_{7}\): Shifted Schwefef’s & [\(-\)100,100] & 10,30,50,100,200,300,400 & \(-\)450 \\ \(F_{8}\): Generalized [PERSON]’s function & [\(-\)100,100] & 10,30,50,100,200,300,400 & \(-\)450 \\ \hline \multicolumn{4}{c}{_Multimodal functions_} & 0 \\ \(F_{9}\): Generalized Rastring’s function & [\(-\)512,512] & 10,30,50,100,200,300,400 & 0 \\ \(F_{10}\): Ackley’s function & [\(-\)32,32] & 10,30,50,100,200,300,400 & 0 \\ \(F_{11}\): Generalized Grievan function & [\(-\)60,60] & 10,30,50,100,200,300,400 & 0 \\ \(F_{12}\): Generalized Penalized 1 function & [\(-\)50,50] & 10,30,50,100,200,300,400 & 0 \\ \(F_{13}\): Alpine function & [\(-\)10,10] & 10,30,50,100,200,300,400 & 0 \\ \(F_{14}\): Shifted Rosenbrock’s function & [\(-\)100,100] & 10,30,50,100,200,300,400 & 390 \\ \(F_{15}\): Shifted Rastring’s function & [\(-\)5,5] & 10,30,50,100,200,300,400 & \(-\)330 \\ \(F_{16}\): Expanded extended Grievan’s plus & [\(-\)3,1] & 10,30,50,100,200,300,400 & \(-\)130 \\ \hline \end{tabular}
\end{table} 
Table 2: Benchmark functions. \begin{table}
\begin{tabular}{c c} \hline
**Algorithm** & **Parameter** & **Value** \\ \hline CGWO & Number of grey wolves & 50 (dim = 10, 30, 50, 100, 200, 300, 400) \\ \(\alpha\) & Linearly decreased from 2 to 0 \\  & Scaling factor (\(F\)) & 0.1 \\  & Max iteration & 500 \\  & Stopping criteria & Max iteration \\ \hline DGWO & Number of grey wolves & 50 (dim = 10, 30, 50, 100, 200, 300, 400) \\  & \(\alpha\) & Linearly decreased from 2 to 0 \\  & Scaling factor (\(F\)) & 0.1 \\  & Max iteration & 500 \\  & Stopping criteria & Max iteration \\ \hline GWO & Number of grey wolves & 50 (dim = 10, 30, 50, 100, 200, 300, 400) \\  & \(\alpha\) & Linearly decreased from 2 to 0 \\  & Max iteration & 500 \\  & Stopping criteria & Max iteration \\ \hline GGSA & Number of particles & 50 (dim = 10, 30, 50, 100, 200, 300, 400) \\  & \(c_{2}^{j}\) & \((-2t^{3}/T^{3})+2\) \\  & \(c_{2}^{j}\) & \((2t^{3}/T^{3})\) \\  & \(G_{0}\) & 1 \\  & \(\alpha\) & 20 \\  & Max iteration & 500 \\  & Stopping criteria & Max iteration \\ \hline ABC & Number of bees & 50 (dim = 10, 30, 50, 100, 200, 300, 400) \\  & Limit & dim \\  & Max iteration & 500 \\  & Stopping criteria & Max iteration \\ \hline \end{tabular}
\end{table} 
Table 3: Initial parameters of algorithms (where \(T\) indicates the maximum number of interactions; \(t\) is the current iteration). +++ ==ERROR: No output for this page==