# A Novel Complex-Valued Encoding Grey Wolf Optimization Algorithm
## 3 Complex-Valued Encoding Grey Wolf Optimization Algorithm (CGWO)
### CGWO Algorithm  
Based on the above analysis, complex-valued encoding, which can be considered as a valid global optimization strategy, is applied to the grey wolf optimizer. The real and imaginary parts of complex numbers are updated independently due to two-dimensional properties of complex numbers. The strategy greatly expands the amount of information contained in the individual gene and enhances the diversity of individual populations. In addition, to improve the local search ability of the algorithm, the differential evolution strategy \"DE/best/2\" [22] is introduced. Under the circumstances, CGWO has the potential to balance global and local searches. In the following section, various benchmark functions are employed to explore the effectiveness of CGWO. The pseudo code of CGWO (Algorithm 2) is as follows:```
1. Initialize the grey wolf population: Let \(\rho_{L}=\left[0,\,\dfrac{B_{L}-A_{L}}{2}\right],\,\,\,\theta_{L}=\left[-2\pi,\,2 \pi\right],\,\,L=1,\,2,\,...\,,\,M\)
2. Get the real and imaginary part of the complex number by Equation (8)
3. Convert to real variables by Equations (15) and (16)
4. **For** all \(X_{i}\)**do**
5. Calculate fitness \(F(X_{i})\)
6. **End for**
7. Get the first three best wolves as \(X_{\alpha}\), \(X_{\beta}\), and \(X_{\delta}\)
8. **While** (\(t<Max\ number\ of\ iterations\))
9. **For** each search agent
10. Update the real part of the current search agent by Equations (9-11)
11. Update the imaginary part of the current search agent by Equations (12-14)
12. **End for**
13. Modify the real and imaginary part by \"DE/best/2\"
14. Select randomly \(r_{2}\
eq r_{2}\
eq r_{3}\
eq r_{4}\
eq i\)
15. \(\overset{\rightarrow}{X}_{new}=\overset{\rightarrow}{X}_{\alpha}+F*(\overset {\rightarrow}{X}_{r1}-\overset{\rightarrow}{X}_{r2})+F*(\overset{ \rightarrow}{X}_{r3}-\overset{\rightarrow}{X}_{r4})\)
16. Update \(a\), \(A\), and \(C\)
17. Convert to real variables by Equations (15) and (16)
18. **For** all \(X_{i}\)**do**
19. Calculate fitness \(F(X_{i})\)
20. **End for**
21. Update \(X_{\alpha}\), \(X_{\beta}\), and \(X_{\delta}\)
22. \(t=t+1\)
23. **End while**
24. Return \(X_{\alpha}\)
```

**Algorithm 2**CGWO algorithm

## 4 Experimental Results and Discussion

### Simulation Platform

All algorithms are tested in Matlab R2012a (7.14), and experiments are executed on AMD Athlon (tm) II X4 640 Processor 3.00 GHz PC with 3G RAM. Operating system is Windows 7. ### Benchmark Functions

To evaluate the performance of the proposed method, termed CGWO, 16 standard benchmark functions are used to demonstrate the performance of the algorithm [23; 24; 25]. They can be divided into two groups: unimodal and multimodal functions. Table 1 lists the functions, where Dim shows dimension of the function, Range is the boundary of search space of the function, and \(f_{\text{min}}\) is the minimum of the function. We compare ABC [7], GGSA [9], CGWO, GWO [16] on benchmark functions to verify the effectiveness of these algorithms dealing with problems of different scales. Moreover, in order to prove the superiority of CGWO over GWO, we derive the functions from complex-valued encoding rather than the differential evolution strategy \"DE/best/2\". CGWO is also compared to the basic GWO with a differential evolution strategy called differential evolution GWO (DGWO). The five algorithms have some parameters that are shown in Table 2.

The experimental initial parameters of the five algorithms are provided in Table 3, the experimental results are provided in Tables 4 and 5. The results are averaged over 20 consecutive experiments. The best results are demonstrated in bold type. The average (AVE) and standard deviation (STD) of the best solution obtained in the last iteration are reported in the form of AVE \(\pm\) STD. Note that the Matlab code of the GGSA algorithm is given in [26]. To improve the performance evaluation of evolutionary algorithms, statistical tests should be conducted [27].