## 2 Basic Theory
### IGTO Performance Evaluation  
Five algorithms--Grey Wolf Optimization [35] (GWO), Particle Swarm Optimization [36] (PSO), GTO [37], IGTO, and Chernobyl Disaster Optimizer [38] (CDO)--were evaluated using single-modal, multi-modal, and unconstrained optimization functions to assess the performance of IGTO in terms of convergence rate, problem-solving capability, and stability. The details of the test functions, including their dimensions, search scope, and theoretical minimum values, are presented in Table 1, while the convergence curves are illustrated in Figure 3.  
As illustrated in Figure 3 above, among the five test functions evaluated, the IGTO algorithm achieves a relatively modest fitness value within the initial 400 iterations. For instance, in the F8 test function, approximately 100 iterations are required to approach the optimal fitness value, whereas other algorithms do not demonstrate a significant reduction in fitness values. The final fitness value attained by the IGTO algorithm is considerably lower than those of the comparative algorithms, with the F5 and F4 test functions achieving reductions to \(10^{-300}\) and \(10^{-200}\), respectively, thereby indicating its superior problem-solving capability. Furthermore, the convergence curves of the IGTO algorithm across the test functions exhibit a high degree of smoothness, with minimal fluctuations observed in the F3 test function, which underscores its stability. In conclusion, the IGTO algorithm demonstrates rapid convergence, robust problem-solving abilities, and consistent results \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Type** & **Function** & **Dim** & **Range** & \(F_{min}\) \\ \hline \multirow{3}{*}{UM} & \(F_{1}\left(x\right)=\sum\limits_{d}^{i=1}x_{i}^{2}\) & 30 & \([-100,100]\) & 0 \\  & \(F_{3}\left(x\right)=max\left(\mid x_{1}\mid,\mid x_{2}\mid,\ldots,\mid\right. \right.\) & 30 & \([-100,100]\) & 0 \\  & \(x_{d}\mid\left.\right)\) & 30 & \([-100,100]\) & 0 \\ \hline \multirow{3}{*}{MM} & \(F_{4}\left(x\right)=\sum\limits_{d}^{i=1}\ (\sum\limits_{i}^{j=1}x_{i})^{2}\) & 30 & \([-100,100]\) & 0 \\  & \(F_{5}\left(x\right)=\sum\limits_{d}^{i=1}\ \mid\mathcal{X}_{i}\ \mid^{i+1}\) & 30 & \([-1,1]\) & 0 \\ \hline \multirow{3}{*}{FM} & \(F_{8}(x)=\sum\limits_{d}^{i=1}\ (\mid x_{i}\mid)+\prod\limits_{d}^{i=1} (\mid x_{i}\mid)\) & 30 & \([-10,10]\) & 0 \\  & \(F_{10}\left(x\right)=\sum\limits_{d}^{i=1}i\cdot\mid x_{i}\mid^{4}\) & 30 & \([-1.28,1.28]\) & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Benchmark functions. Figure 2: Flowchart of IGTO. in high-dimensional complex optimization scenarios. It is also well-suited for diagnosing rolling bearing faults, which necessitate high levels of real-time optimization and result stability. To better illustrate the optimization performance of IGTO, two metrics--average fitness value (Avg) and standard deviation (Std)--are employed to evaluate the proposed algorithm. The average fitness value intuitively reflects the convergence accuracy and search capability of the algorithm, while the standard deviation indicates the degree of deviation between experimental results and the average value. The optimization results of different algorithms on the benchmark test functions are presented in Table 2 as follows. As shown in Table 2, IGTO stands out among various optimization algorithms due to its superior stability and precision. Compared with other algorithms, IGTO consistently achieves the theoretical minimum, whereas others still exhibit considerable errors across multiple functions. This highlights IGTO's remarkable advantage in high-precision optimization tasks. Furthermore, IGTO maintains extremely low fluctuations during the