## 3 Experimental Results and Analysis  
HHO [13], SSA [16], PSO [24], Gold-SA [25], and artificial gorilla troops optimizer (GTO) [26] are selected as the comparison algorithms to verify the search performance of ISSA. The population size of all algorithms is set to 30, and the number of iterations is set to 500. In the ISSA proposed in this paper, the ratio of generators is 0.6, the ratio of discoverers is 0.7, and the ratio of vigilantes is 0.2. Meanwhile, Gaussian perturbation is introduced when the global optimum value is below the threshold 1.00 \(\times\) 10\({}^{-10}\) in two consecutive iterations. The parameters of the other algorithms are consistent with their literature. The whole experiment is divided into four parts: convergence accuracy analysis, convergence speed analysis, rank sum test, and complexity analysis. The simulation experiment described in this article was conducted on a Windows 11 64-bit operating system with an AMD Ryzen 7 5800H processor with Radeon Graphics 3.20 GHz, 16 GB of RAM, and MATLAB 2014B. All experimental results are the average values after 30 runs. In this section, 23 test functions are selected from CEC's benchmark functions, which are commonly used for optimization testing of meta-heuristic algorithms [16; 27; 28; 29]. The functions \(F_{1}\)-\(F_{7}\) are unimodal functions, the functions \(F_{8}\)-\(F_{13}\) are multimodal functions, and the functions \(F_{14}\)-\(F_{23}\) are fixed-dimension functions. Table 1 lists the parameters of each function, including its expression, dimension, range, and optimal value. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Name** & **Formula of Functions** & **Dim** & **Range** & **Best** \\ \hline Sphere & \(F_{1}(x)=\sum_{i=1}^{n}x_{i}^{2}\) & 30 & [100, 100] & 0 \\ Schwefel 2.22 & \(F_{2}(x)=\sum_{i=1}^{n}[x_{i}+\prod_{i=1}^{n}[x_{i}]]\) & 30 & [10, 10] & 0 \\ Schwefel 1.2 & \(F_{2}(x)=\sum_{i=1}^{n}[(x_{i})^{2}-x_{i})^{2}\) & 30 & [100, 100] & 0 \\ \hline Schwefel 2.21 & \(F_{4}(x)=\max_{i=1}([x_{i}],1\leq i\leq n)\) & 30 & [100, 100] & 0 \\ Rosenbrock & \(F_{5}(x)=\sum_{i=1}^{n}[100\times(x_{i}-x_{i}^{2})+(x_{i}-1)^{2}]\) & 30 & [30, 30] & 0 \\ Sep & \(F_{6}(x)=\sum_{i=1}^{n}[(x_{i}+0.5)^{2}]\) & 30 & [100, 100] & 0 \\ Quartic & \(P_{7}(x)=x_{7}^{4}+random(0,1)\) & 30 & [128, 128] & 0 \\ \hline Schwefel 2.26 & \(F_{8}(x)=\sum_{i=1}^{n}x_{i}\sin(\sqrt{|x_{1}|})\) & 30 & [500, 500] & \(-418.9829\times n\) \\ Rastrigin & \(F_{9}(x)=\sum_{i=1}^{n}[x_{i}^{2}-10\cos(2\pi x_{i})+10]\) & 30 & [512, 512] & 0 \\ Ackley & \(F_{10}(x)=-20\exp\left(-20\sqrt{\frac{1}{2}\sum_{i=1}^{n}x_{i}^{2}}\right)- \exp\left(\frac{1}{2}\sum_{i=1}^{n}\cos(2\pi x_{i})\right)+20+\epsilon\) & 30 & [32, 32] & 0 \\ [PERSON] & \(F_{11}(x)=\frac{10}{2}\sum_{i=1}^{n}\sum_{i=1}^{n}-11\sin(\sqrt{|x_{i}|})+1\) & 30 & [600, 600] & 0 \\  & \(F_{12}(x)=\frac{1}{2}\left\{10\sin^{2}(\pi y_{1})+\sum_{i=1}^{n}-1\left(y_{i} -1\right)^{2}+10\sin^{2}(\pi y_{1})+\sum_{i=1}^{n}\left(y_{i},100,4\right) \right.\) & & \\ Penalized 1 & \(y_{i}=1+\frac{x_{i}^{2}}{4}\) & \(x=(y_{i},x_{i},x_{i},x_{i})=\left\{ \begin{array}{ll}x(x-x_{i})+\epsilon\\ -\epsilon<x_{i}\end{array} \right.\) & 30 & [50, 50] & 0 \\  & \(\left.\left(x-x_{i}\right)^{n}\right.\) & \(x_{i}<a\) & & \\ Penalized 2 & \(F_{13}(x)=0.1\left[\sin^{2}(3\pi x_{i})+\sum_{i=1}^{n}\left(x_{i}-1\right)^{ 2}\left[1+\sin^{2}(3\pi x_{i+1})+(x_{i}-1)^{2}(1+\sin^{2}(2\pi x_{i+1}))\right] \right.\) & & \\  & \(\left.+\sum_{i=1}^{n}\left(x_{i},100,4\right)\right.\) & & & \\ \hline Fooles & \(F_{13}(