## 4 Experimental Setup
### Benchmark Test Functions  
To comprehensively evaluate the convergence behavior and global optimization performance of the proposed IGWOHHO algorithm, six representative benchmark functions were selected. Schwefel 2.22 and Rosenbrock are unimodal functions, used to assess the conver  
Figure 8: Main effect chart of S/N ratio for algorithm parameter evaluation. Figure 7: Sensitivity analysis of algorithm parameters. gence speed and accuracy of the algorithm. Ackley and Penalized are multimodal functions characterized by complex landscapes with numerous local optima, making them suitable for testing global exploration capabilities. Foxholes and Kowalik are fixed-dimensional multimodal functions with intricate topologies, commonly used to evaluate local exploitation ability and solution stability. The definitions, dimensional settings, variable bounds, and known global optima for these benchmark functions are consistent with standard references in the literature [49; 50], ensuring a reliable basis for performance comparison. The mathematical formulations are provided in Equations (55)-(60). The Schwefel 2.22 function, defined in Equation (55), is a unimodal and separable test function that evaluates performance based on the combination of the sum and product of the absolute values of decision variables. \[F_{1}(x)=\sum_{i=1}^{n}\lvert x_{i}\rvert+\prod_{i=1}^{n}\lvert x_{i}\rvert, \tag{55}\]  
where \(x=(x_{1},\,x_{2},\,\ldots,\,x_{n})\in[-10,\,10]^{n}\), and the global minimum is 0, which occurs when \(x_{i}=0\). The Rosenbrock function, defined in Equation (56), is a unimodal and non-separable function characterized by a narrow, curved valley. It is commonly used to evaluate an algorithm's convergence ability. \[F_{2}(x)=\sum_{i=1}^{n-1}\left[100\left(x_{i+1}-x_{i}^{2}\right)^{2}+(x_{i}-1) ^{2}\right], \tag{56}\]  
where \(x\in[-30,\,30]^{n}\), and the global minimum is 0, which occurs when \(x_{i}=1\). The Ackley function, defined in Equation (57), is a multimodal and non-separable function commonly used to evaluate global exploration capability in complex search spaces. \[F_{3}(x)=-20 exp\left(-0.2\sqrt{\frac{1}{n}\sum_{i=1}^{n}x_{i}^{2}}\right)-exp \left(\frac{1}{n}\sum_{i=1}^{n}cos(2\pi x_{i})\right)+20+e, \tag{57}\]  
where \(x\in[-32,\,32]^{n}\), and \(e\) is Euler's number. The global optimum is 0. The Penalized function, defined in Equation (58), incorporates penalty terms that increase the complexity of the optimization process, making it suitable for evaluating an algorithm's robustness. \[F_{4}(x)=\frac{\pi}{n}\Bigg{\{}10 sin(\pi y_{1})+\sum_{i=1}^{n-1}\left(y_{i}-1 \right)^{2}\left[1+10 sin^{2}(\pi y_{i+1})\right]+\left(y_{n}-1\right)^{2} \Bigg{\}}+\sum_{i=1}^{n}u(x_{i},10,100,4), \tag{58}\]  
where \(y_{i}=1+u(x_{i})\). The function introduces penalty terms for solutions outside the allowed region and has a global minimum value of 0 within the domain \(x\in[-50,\,50]^{n}\). The Foxholes function, defined in Equation (59), features a highly complex landscape with numerous local optima. It is commonly used to evaluate an algorithm's ability to escape local optima and perform an effective global search. \[F_{5}(x)=\left[\frac{1}{500}+\sum_{j=1}^{25}\frac{1}{j+\sum_{l=1}^{2}\left(x _{i}-a_{ij}\right)^{6}}\right]^{-1}, \tag{59}\]  
where \(x\in[-65,\,65]^{2}\), and \(a_{ij}\) are constants define the positions of the holes in the landscape. The global minimum is approximately 0.998. The Kowalik function, defined in Equation (60), is a low-dimensional multimodal function commonly used to evaluate local search accuracy and fine-tuning capability. \[F_{6}(x)=\sum_{i=1}^{11}\left[a_{i}-\frac{x_{1}\big{(}b_{i}^{2}+b_{i}x_{1}\big{)}}{ b_{i}^{2}+b_{i}x_{3}+b_{i}x_{4}}\right]^{2}, \tag{60}\]  
where \(x=(x_{1},\,x_{2},\,\ldots,\,x_{n})\)\(\in\)[\(-5,\,5\)]\({}^{4}\), and the parameters \(a_{i}\) and \(b_{i}\) are constants typically derived from empirical datasets. The global minimum is 0.