## 4 Analysis of Experimental Results
### Simulation Experiment
#### 4.1.2 Comparative Analysis of Performance with Other Algorithms  
In order to verify the overall performance of the IGEO, eight algorithms were selected for comparison, including the butterfly optimization algorithm (BOA) [13], the grey wolf optimizer (GWO) [10], particle swarm optimization (PSO) [4], the sine cosine algorithm (SCA) [52], the slap swarm algorithm (SSA) [12], the whale optimization algorithm (WOA) [9], the golden eagle optimizer algorithm (GEO) [26], and the golden eagle optimizer with double learning strategies (GEO_DLS) [42]. To ensure fairness, the population number for all algorithms was 50, and the maximum number of iterations was 1000. In addition, the parameters of each original algorithm were the best, as shown in Table 3.  
Table 4 displays the analytical results in relation to the other eight algorithms, along with the mean, standard deviation (std), and average running time. These results show the accuracy of the convergence and optimization of the algorithms. From the results for mean value and standard deviation, it can be seen that IGEO has the best search accuracy: the IGEO was able to readily obtain an excellent advantage that could be matched by no other algorithm on F1, F2, F4, F5, F7, F8, and F11, and the only other algorithms that achieved the same grades as IGEO were GEO_DLS on F3, WOA on F6 and F9, GWO on F10, and PSO on F12. The performance on the continuous unimodal functions F1-F2 demonstrate the superior optimization capabilities of IGEO. The performance on the challenging multivariate functions (F3-F8) demonstrates IGEO's capacity to avoid local maxima. Only WOA on F6 and GEO_DLS on F3 have an optimization precision as good as IGEO. Last but not least, IGEO's performance on the fixed low-dimensional functions P9-F12 demonstrates its all-encompassing capabilities. Unfortunately, IGEO's performance in this area is marginally subpar: the optimization accuracy of WOA on F9, GWO on F10, and PSO on F12 are all equal to that of IGEO. Due to the characteristics of the method itself, GEO, GEO_DLS, and IGEO have the longest mean running times, while IGEO performs the best overall. An MAE ranking [53] on the best outcomes of 50 experiments was performed in this work in order to more accurately assess the optimization performance of each method on the test function. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Fun No.** & **Name** & **Equation** & **Features** & **D** & **Bounds** & **f*** \\ \hline F12 & Levi 13 & \(F_{12}(x)=sin^{2}(3\pi x_{1})+(x_{1}-1)^{2}\Big{(}1+sin^{2}(3\pi x_{2})\Big{)}\) & Fixed low-dim & 2 & \([-10,10]^{\text{D}}\) & 0 \\  & & \(+(x_{2}-1)^{2}\Big{(}1+sin^{2}(2\pi x_{2})\Big{)}\) & & & & \\  & & \(\mathbf{f^{*}}\) represents the theoretical optimal value of each test function. & & & & \\  & & D represents the dimension of the test function. & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: _Cont._ \begin{table}
\begin{tabular}{c c} \hline \hline
**Algorithms** & **Parameter Settings** \\ \hline BOA & \(p=0.8\), \(c=0.01\), \(a=0.1\) \\ GWO & \(a_{max}=2\), \(a_{min}=0\) \\ PSO & \(\omega_{max}=0.9\), \(\omega_{min}=0.2\), \(c_{1}=c_{2}=2\) \\ SCA & \(a=2\) \\ SSA & - \\ VOA & \(b=1\) \\ GEO & \(P_{c}=[P_{c}^{0}\ P_{c}^{T}]=[1.05]\), \(P_{a}=[P_{a}^{0}\ P_{c}^{T}]=[0.5\ 2]\) \\ GEO\_DLS & \(P_{c}=[P_{c}^{0}\ P_{c}^{T}]=[10.5]\), \(P_{a}=[P_{a}^{0}\ P_{c}^{T}]=[0.5\ 2]\), \(\epsilon=3\), \(\mu\)\(\in\)[0,1], Q\(\in\)[0,1] \\ IGEO & \(\omega_{\text{ini}}=0.9\), \(\omega_{end}=0.4\), \(P_{c}=[P_{c}^{0}\ P_{c}^{T}]=[10.5]\), \(P_{a}=[P_{a}^{0}\ P_{a}^{T}]=[0.5\ 2]\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameter settings of each algorithm. In a quantitative analysis of all algorithms, the average absolute error of 12 benchmark functions is used to rank all algorithms. The algorithm's performance increases as MAE decreases. The formula for determining the MAE ranking of these benchmark functions is given in Equation (16), and Table 5 displays the ranking of these algorithms.