## 2 Basic Theory
### The Improved Gorilla Troops Optimizer
#### 2.2.2 Improved Artificial Gorilla Tropp Algorithm
For each dimension \(j\) (where \(j=1\), \(2\),..., D), the inverse solution \(GX^{*}_{j}\) is computed as follows:  
\[GX^{*}_{j}=\frac{lb_{j}+ub_{j}}{2}+\frac{lb_{j}+ub_{j}}{2n}-\frac{GX_{j}}{n} \tag{14}\]  
where \(lb_{j}\) and \(ub_{j}\) is the lower and upper bounds of the \(j\)-th dimension; \(GX_{j}\) is the current candidate solution for the \(j\)-th dimension; \(GX^{*}_{j}\) is the inverse solution for the \(j\)-th dimension. Figure 1: Results of the initial population. In the process of generating a new inverse solution, it is essential to evaluate the fitness values of both the inverse solution and the candidate solutions. The solution exhibiting the higher fitness value is subsequently chosen for the optimization process outlined below:  
\[GX_{\text{next}}=\left\{ \begin{matrix}GX^{*},\text{if }F\left(GX^{*}\right)<F \left(GX\right)\\ GX,\text{ otherwise}\end{matrix} \right. \tag{15}\]  
where \(GX_{\text{next}}\) is the selected solution; \(F\left(\cdot\right)\) is the fitness function of the problem. In the exploration phase of GTO, the LOBL mechanism enhances the algorithm by updating inverse solutions. This approach effectively mitigates the risk of the algorithm becoming ensnared in local optima, thereby improving the precision of local exploitation. 3A\(\beta\)HC Global Solution Update  
In the context of GTO, the exploitation phase frequently encounters challenges when addressing high-dimensional and complex optimization problems, often resulting in suboptimal solutions. To mitigate this issue, A\(\beta\)HC has been integrated into the GTO framework, facilitating a transition between global and local search strategies. During the initial phases of the search, a larger step size is employed to swiftly navigate the solution space and identify promising high-quality solutions. Conversely, in the later stages, the dynamic modification of the \(\beta\) value serves to avert premature convergence within a localized region, thereby ensuring both the quality of the solution and the thoroughness of the optimization process [34]. A\(\beta\)HC conducts local search utilizing two control operations: the N-operator and the \(\beta\)-operator. The corresponding mathematical models are delineated as follows:  
The N-operator generates neighborhood solutions:  
\[X_{i}^{\prime}=\left(x_{i1}^{\prime},x_{i2}^{\prime},\ldots,x_{iD}^{\prime}\right) \tag{16}\]  
where \(x_{ij}^{\prime}\) is the \(j\)-th dimension of the solution generated through perturbation. The \(\beta\)-operator dynamically adjusts the step size:  
\[\beta\left(t\right)=\beta_{min}+\left(\beta_{max}-\beta_{min}\right)\times \frac{t}{MaxIter} \tag{17}\]  
where \(\beta_{min}\), \(\beta_{max}\) is the minimum and maximum step size values, respectively; \(t\) is the current iteration number; \(MaxIter\) is the maximum number of iterations. Compared to conventional solution update mechanisms, A\(\beta\)HC mitigates early convergence and improves the convergence speed of GTO, thereby facilitating a more rapid attainment of optimal or near-optimal solutions. 4 Summary  
In summary, the IGTO is developed by incorporating Circle Chaotic Mapping, LOBL, and A\(\beta\)HC as the fundamental mechanisms for initialization, local solution updates, and global solution updates, respectively. The integration of these enhancement mechanisms is designed to improve the adaptability and robustness of IGTO in addressing multimodal, high-dimensional, and complex optimization problems. The implementation process is illustrated in Figure 2 below.