## 5 Experiment for Benchmark Functions  
The CEC2014 benchmark function is used for comparing the performance of different algorithms on a unified standard in this section and the ten typical benchmark functions listed in Table 1 were used to evaluate the proposed HSCSO algorithm search performance. Among of these benchmark functions, the first five benchmark functions are uni-model test functions and the rest are multi-model. In order to prove the effectiveness of the HSCSO algorithm, five commonly used heuristic algorithms and a modified sand cat warm algorithm were tested for comparison in dimensions of 30 and 100, namely the Multi-Verse Optimizer (MVO) [30], Ant Lion Optimizer (ALO) [31], Sine Cosine Algorithm (SCA) [32], Grey Wolf Optimizer (GWO) [33], Sand Cat Swarm Optimization (SCSO) [17], and Modified Sand Cat Swarm Optimization (MSCSO) [20]. The corresponding parameter of the tested algorithms are given in Table 2, where \(G\) represents the maximum number of iterations, \(NP\) represents the population size, \(WEP\) represents the wormhole existence probability, \(TDR\) represents the travelling distance, \(a\) is the parameter that controls the amplitude of the sine and cosine functions, \(Sm\) represents the maximum sensitivity of the sand cats, \(\sigma\) is the scale factor of the Levy flight step, \(\beta\) is the Levy Exponent, and \(c_{1}\) and \(c_{2}\) are the momentum factor and the bellicose factor of this paper. During the testing process, the iteration will end when the inequality \(|v_{t}-V|\leq\varepsilon=~{}10^{-10}\) is met, where \(v_{t}\) is given the definition of the best value, \(V\) represents the actual optimal value, and \(\varepsilon\) is a symbol used for the convergence precision setting. Meanwhile, three indices (namely min, mean, and std) are introduced which, respectively, represent the exploration ability, the convergence accuracy, and the robustness. To ensure the accuracy of the test results, we tested each algorithm 500 times and took the mean as the result. And, the final testing results are listed in Table 3. These algorithms' convergence trend of the test function f1 \(\sim\) f10 are also shown in Figure 5. Since the convergence trend of different algorithms in the case of 100 dimensions is similar to that of images in 30 dimensions, this paper only exhibits the figures in a 30-dimension condition. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Name** & **Definition** & **Domain** & **Minimum** \\ \hline Sphere & \(\mathrm{F_{1}(x)=\sum_{i=1}^{n}x_{1}^{2}}\) & \([-100,100]\) & \(0\) \\ \hline Schwefel 2.22 & \(\mathrm{F_{2}(x)=\sum_{i=1}^{n}|x_{i}|+\prod_{i=1}^{n}|x_{i}|}\) & \([-10,10]\) & \(0\) \\ \hline Schwefel 1.2 & \(\mathrm{F_{3}(x)=\sum_{i=1}^{n}\left(\sum_{i-1}^{i}x_{i}\right)^{2}}\) & \([-100,100]\) & \(0\) \\ \[PERSON] & \(\mathrm{F_{4}(x)=\sum_{i=1}^{n-1}\left[100\left(x_{i+1}-x_{i}^{2}\right)^{2}+ \left(x_{i}-1\right)^{2}\right]}\) & \([-30,30]\) & \(0\) \\ \[PERSON] & \(\mathrm{F_{5}(x)=\sum_{i=1}^{n}([x_{i}+0.5])^{2}}\) & \([-100,100]\) & \(0\) \\ \[PERSON] & \(\mathrm{F_{6}(x)=\sum_{i=1}^{n}\left(x_{i}^{2}-10\mathrm{cos}(2\pi x_{i})+10 \right)}\) & \([-5.12,5.12]\) & \(0\) \\ \[PERSON] & \(\mathrm{F_{7}(x)=-20\mathrm{exp}\left(-0.2\sqrt{\frac{1}{n}\sum_{i=1}^{n}x_{i}^ {2}}\right)-\mathrm{exp}\left(\frac{1}{n}\sum_{i=1}^{n}\mathrm{cos}(2\pi x_{i} )\right)+20+e}\) & \([-32,32]\) & \(0\) \\ \hline Griewank & \(\mathrm{F_{8}(x)=\frac{1}{400\mathrm{e}}\sum_{i=1}^{n}x_{i}^{2}-\prod_{i=1}^{n} \mathrm{cos}\left(\frac{x_{i}}{\sqrt{i}}\right)+1}\) & \([-600,600]\) & \(0\) \\ \hline Penalized1 & \(\mathrm{F_{9}(x)=\frac{\pi}{n}\left\{10\mathrm{sin}(\pi y_{1})+\sum_{i=1}^{n-1 }\left(y_{i}-1\right)^{2}\left[1+10\mathrm{sin}^{2}\left(\pi y_{i+1}\right) \right]+\left(y_{n}-1\right)^{2}\right\}+}\) & \([-50,50]\) & \(0\) \\ \hline Penalized2 & \(\mathrm{F_{10}(x)=0.1\left\{\sin^{2}(3\pi x_{1})+\sum_{i=1}^{n}(x_{i}-1)^{2} \left[1+\sin^{2}(3\pi x_{i}+1)\right]+\right.\right.\) & \([-50,50]\) & \(0\) \\ \hline \hline \end{tabular}
\end{table}