## 2 Basic Theory
### The Improved Gorilla Troops Optimizer
#### 2.2.2 Improved Artificial Gorilla Tropp Algorithm  
Although the GTO algorithm presents several advantages in the context of global optimization problems, it also reveals significant limitations when tackling complex optimization tasks. Initially, the population in GTO is generated randomly, resulting in reduced diversity within the early population. Secondly, the exploration phase in GTO is predominantly dependent on random migration. Lastly, the application of GTO to high-dimensional and complex problems is often hindered by premature convergence. In order to address these deficiencies, the following enhancements are proposed for the GTO:  
\(\lx@paragraphsign\) Initialization of the Circle Chaotic Mapping  
In GTO, the initial population is generated through a random search process, which often results in a population that inadequately represents the solution space, consequently constraining the algorithm's exploratory capabilities. The traversal characteristics of Chaotic Mapping facilitate a more extensive coverage of the solution space, thereby improving the diversity of the initial population. Among the various chaotic mapping techniques, Circle Chaotic Mapping has been selected as the enhancement method in this study due to its superior exploratory performance. Research has demonstrated that Circle Chaotic Mapping surpasses both Logistic and Tent chaotic mappings in terms of exploration efficacy [32]. The mathematical representation of Circle Chaotic Mapping is as follows:  
\[z_{k+1}=z_{k}+\frac{b-a}{2\pi}\cdot\sin\;(2\pi z_{k})\text{mod }(1),z_{k}\in\;(0,1) \tag{11}\]  
where \(z_{k}\) is the value at the current step; \(z_{k+1}\) is the value at the next step; \(a=0.5,b=0.2\) are constants defining the chaotic behavior. In the study, both random search and Circle Chaotic Mapping were independently executed 300 times in a two-dimensional space to compare their initialization effects. The results are shown in Figure 1 below. Figure 1 illustrates that the traversal of Circle Chaotic Mapping exhibits a more extensive and uniform distribution within the feasible domain of [0, 1]. In comparison to random search methods, Circle Chaotic Mapping encompasses a larger segment of the solution space and achieves a more equitable distribution. 2 LOBL Partial Solution Update  
LOBL integrates the conventional Opposition-Based Learning (OBL) methodology with the principles of convex lens imaging [33]. The fundamental concept of LOBL involves the simultaneous computation and comparison of candidate solutions alongside their corresponding inverse solutions, subsequently selecting the superior solution for the next iteration. Consequently, in this study, LOBL is employed during the exploration phase to broaden the search range and assist the algorithm in escaping from local optima. The phenomenon known as OBL pertains to the formation of a reduced, inverted image of an object when the object is positioned at a distance greater than twice the focal length from the lens. Utilizing the principles of lens imaging and the properties of similar triangles, the following geometric relationship can be derived:  
\[\frac{(lb+ub)}{2}-GX=\frac{GX^{*}-(lb+ub)}{2}=h/h^{*} \tag{12}\]  
Utilizing the aforementioned formula, let the ratio factor be denoted as \(n=h/h^{*}\). The inverse solution \(GX^{*}\) can be computed as follows:  
\[GX^{*}=\frac{lb+ub}{2}+\frac{lb+ub}{2n}-\frac{GX}{n} \tag{13}\]  
The OBL strategy represents a specific instance of the LOBL approach. In contrast to OBL, LOBL possesses the capability to dynamically modify the inverse solution by altering the ratio factor \(n\), thereby providing an expanded search range. Algorithm optimization is generally conducted within multi-dimensional spaces; consequently, we extend LOBL to a D-dimensional framework.