## 3 Mixed-Strategy Harris Hawk Optimization
### Gaussian Random Walk Strategy
In this benchmark function set, 12 benchmark functions are selected to test and research the performance of MSHHO, including 7 unimodal benchmark function problems (C1 to C7) and 5 multimodal benchmark function problems (C8 to C12). A unimodal benchmark function indicates that there is only one extreme point in a given range of seek space, while a multimodal benchmark function indicates that there are multiple extreme points in a given range of seek space. The unimodal benchmark functions and multimodal benchmark functions used are displayed in Tables 2 and 3. \begin{table}
\begin{tabular}{c c c} \hline
**ID** & **Parameters and Values** \\ \hline \#All & Population size: 30; Maximum iterations: 500; Independent runs: 30 \\ DE & Mutation operator at interval: [0.4, 1] \\ GA & Cross probability: \(ppc=0.8\); Mutation probability: \(ppm=0.05\) \\ WOA & Probability of encircling mechanism = 0.5; Spiral factor = 1 \\ CDO & Alpha speed: 16,000 km/s; Beta speed: 270,000 km/s; Gamma speed: 300,000 km/s \\ PSO & The weight of inertia reduces linearly at interval: [0.2, 0.9] \\ OMA & Objective parameter: 1.40; Eyepiece parameters: 0.55 \\ AOA & Sensitive parameter: \(a=5\); Control parameter: \(u=0.5\) \\ DA & \(w=[0.2,0.9]\); \(s1=0.1\); \(a1=0.1\); \(c1=0.7\); \(f1=1\); \(e1=1\) \\ GWO & Convergence parameter at interval: \([0,2]\) \\ HHO & \(En\) decreased linearly at interval: \([-1,1]\) \\ MSHHO & \(En\) decreased nonlinearly at interval: \([-1,1]\) \\ \hline \end{tabular}
\end{table} 
Table 1: Algorithm parameters. \begin{table}
\begin{tabular}{c c c c c} \hline  & **Function** & **Dim** & **Interval** & **Target** \\ \hline \(C1(x)=\sum\limits_{k=1}^{D}x_{k}^{2}\) & 30 & \([-100,100]\) & 0 \\ \(C2(x)=\sum\limits_{k=1}^{D}|x_{k}|+\prod\limits_{k=1}^{D}|x_{k}|\) & 30 & \([-10,10]\) & 0 \\ \(C3(x)=\sum\limits_{k=1}^{D}\left(\sum\limits_{j=1}^{k}x_{j}\right)^{2}\) & 30 & \([-100,100]\) & 0 \\ \(C4(x)=max\left\{|x_{k}|,1\leq k\leq D\right\}\) & 30 & \([-100,100]\) & 0 \\ \(C5(x)=\sum\limits_{k=1}^{D-1}\left[100\big{(}x_{k+1}-x_{k}^{2}\big{)}^{2}+(x_ {k}-1)^{2}\right]\) & 30 & \([-100,100]\) & 0 \\ \(C6(x)=\sum\limits_{k=1}^{D}\left([x_{k}+0.5]\right)^{2}\) & 30 & \([-30,30]\) & 0 \\ \(C7(x)=\sum\limits_{k=1}^{D}kx_{k}^{4}+random[0,\;1)\) & 30 & \([-1.28,1.28]\) & 0 \\ \hline \end{tabular}
\end{table} 
Table 2: Monomodal benchmark functions (C stands for CEC2005-F). All the experiments were carried out on Matlab R2021a. The computer was a 64-bit operating system, Windows S10 was a home Chinese version, the RAM was 16 GB, and the processor was Intel (R) Core (TM) i7-9750h (2.60 GHz). To testify to the effectiveness of MSHHO, 12 typical functions in the CEC2005 benchmark function set were selected to compare and test MSHHO with 10 popular algorithms, and the test results were analyzed. The maximum number of iterations was 500, and the population size was 30. ### Qualitative Analysis for MSHHO

#### 4.2.1 Convergence Behavior Analysis

To confirm MSHHO's convergence ability, its behaviors are shown in Figure 7, with six figures based on different benchmark functions. In the first column, the 3D structures of the test functions are displayed. In the second column, the final positions of the individuals are clearly shown. Although individuals are spread throughout the search space, they are mainly near the optimal solution. This proves the algorithm's strong exploration and exploitation ability. The third column shows that the average fitness value changes with the number of iterations. Although the initial fitness value is high, it becomes low and stable after 100 iterations, indicating that MSHHO has a rapid convergence capability. The fourth column shows that the agent's trajectory exhibits large fluctuations during the initial iterations, demonstrating MSHHO's effectiveness in avoiding local optima and finding high-quality solutions. The last column shows the MSHHO convergence curve, which appears smooth in most cases. Based on this, MSHHO is proven to have good convergence.