## 3 Mixed-Strategy Harris Hawk Optimization
### Gaussian Random Walk Strategy
\begin{table}
\begin{tabular}{l l l l} \hline \hline
**Function** & **Dim** & **Interval** & **Target** \\ \hline \(\mathsf{C8}(x)=\sum\limits_{i=1}^{D}\big{[}x_{i}^{2}-10 cos(2\pi x_{i})+10\big{]}\) & 30 & \([-5.12,5.12]\) & 0 \\ \(\mathsf{C9}(x)=-20 exp\Bigg{(}-0.2\sqrt{\frac{1}{D}\sum\limits_{i=1}^{D}\ x_{i}^{2}} \bigg{)}-exp\Bigg{(}\frac{1}{D}\sum\limits_{i=1}^{D}\ cos(2\pi x_{i})\Bigg{)}+20 +e\) & 30 & \([-32,32]\) & 0 \\ \(\mathsf{C10}(x)=\frac{1}{4000}\sum\limits_{i=1}^{D}\ x_{i}^{2}-\prod\limits_{i= 1}^{D}\ cos\bigg{(}\frac{x_{i}}{\sqrt{i}}\bigg{)}+1\) & 30 & \([-600,600]\) & 0 \\ \(\mathsf{C11}(x)=\frac{\pi}{D}\Bigg{\{}10 sin(\pi y_{1})+\sum\limits_{i=1}^{D-1 }\ (y_{i}-1)^{2}\Big{[}1+10 sin^{2}(\pi y_{i+1})\Big{]}+(y_{D}-1)^{2}\bigg{\}}\ \ \ +\) & & \\ \(\sum\limits_{i=1}^{D}\ u(x_{i},10,\,100,\,4)\) & 30 & \([-50,50]\) & 0 \\ \(y_{i}=1+\frac{x_{i}+1}{4}\), & 30 & \([-50,50]\) & 0 \\ \(u(x_{i},a,k,m)=\begin{cases}k(x_{i}-a)^{m}&x_{i}>a\\ 0&-a<x_{i}<a\\ k(-x_{i}-a)^{m}&x_{i}<-a\end{cases}\) & \\ \(\mathsf{C12}(x)=\) & 0.1\bigg{\{}sin^{2}(3\pi x_{1})+\sum\limits_{i=1}^{D}\ (x_{i}-1)^{2}\Big{[}1+sin^{2}(3\pi x_{i+1})\Big{]}\) & 30 & \([-50,50]\) & 0 \\  & \(+(x_{D}-1)^{2}\Big{[}1+sin^{2}(\pi x_{D})\Big{]}\Big{\}}+\sum\limits_{i=1}^{D }\ u(x_{i},5,\,100,\,4)\) & & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Multimodal benchmark functions. #### 4.2.2 Population Diversity Analysis

The MSHHO algorithm maintains excellent population diversity. To verify this, (26) and (27) are used for the calculation [54]. \[ppc(t)=\frac{\sum_{i=1}^{N}X_{ij}(t)}{dim} \tag{26}\]

\[ppd(t)=\sqrt{\sum_{i=1}^{N}\sum_{j=1}^{dim}\biggl{(}X_{ij}(t)-\frac{\sum_{i=1}^{ N}X_{ij}(t)}{dim}\biggr{)}^{2}} \tag{27}\]

where \(ppd(t)\) represents the dispersion of the population from the center of mass during the \(t\)th iteration, \(ppc(t)\) indicates population diversity, \(X_{ij}(t)\) denotes the position of the ith agent in the jth dimension at the \(t\)th iteration. Figure 8 shows the experimental results of HHO and MSHHO regarding population diversity. The MSHHO curves fluctuate significantly in the early iterations and then gradually stabilize, while HHO's low population diversity reveals its limitations. These results suggest that MSHHO is more effective in avoiding premature convergence. Figure 7: Convergence behaviors of MSHHO (CEC2005-F can be abbreviated to C). #### 4.2.3 Exploration and Exploitation Analysis

MSHHO divides the search process into two phases: exploration and exploitation. Maintaining a balance between these two phases is crucial. [PERSON] et al. introduced a method to measure and analyze the exploration and exploitation capabilities of an algorithm [55]. The corresponding equations are as follows. \[ddv(t)=\frac{\sum_{j=1}^{dim}\sum_{i=1}^{N}\abs{median\big{(}X_{j}(t)\big{)}-X_ {ij}(t)}}{dim\times N} \tag{28}\]

\[erp(\%)=\frac{ddv(t)}{ddv_{max}}\times 100\% \tag{29}\]

\[etp(\%)=\frac{\abs{ddv(t)-ddv_{max}}}{ddv_{max}}\times 100\% \tag{30}\]

where \(ddv(t)\) represents dimension diversity, \(ddv_{max}\) denotes the maximum diversity, and \(erp(\%)\) and \(etp(\%)\) indicate the exploration and exploitation percentages, respectively. Figure 9 presents the experimental results on exploration and exploitation. For unimodal functions, MSHHO's exploitation rate steadily increases with iterations, whereas HHO shows sudden fluctuations, demonstrating MSHHO's faster convergence. For multimodal functions, which have multiple extreme values and are prone to the local optimal, the initial iteration shows fluctuations in exploration and exploitation rates. This indicates

Figure 8: The population diversity comparison. MSHHO reduces the likelihood of getting trapped in the local optimal. Even when applied to challenge benchmark functions like hybrid problems and composition problems, MSHHO maintains a satisfactory exploration and exploitation rate. Overall, MSHHO effectively balances exploration and exploitation, preserving population diversity.