# Gaslike Social Motility: Optimization Algorithm with Application in Image Thresholding Segmentation
## 3 GSM Algorithm Performance Results  
To evaluate the efficiency of the GSM algorithm, it was implemented and tested on a set of benchmark functions. The GSM algorithm was compared on equal terms with several well-known optimization algorithms, including PSO, DE, BA, ABC, AHA, AHA-AO, CBO, ECBO, and SNS. The benchmark functions were selected to represent a diverse range of problem types, such as unimodal, multimodal, regular, irregular, separable, non-separable, and multidimensional functions. The parameters used for each algorithm in this study are provided in Table 2.  
A unimodal function has a single global optimum with no or only one local optimum, while a multimodal function features multiple local optima. Optimization algorithms are often tested on multimodal functions to assess their ability to avoid local optima. Algorithms with poor exploration capabilities may struggle to search the space thoroughly and may converge to suboptimal solutions. Additionally, flat search spaces pose challenges as they lack gradient information to guide the algorithm toward the global optimum [20]. Both separable and non-separable functions were included in the tests to further evaluate algorithm performance [20]. High-dimensional search spaces present another significant challenge for optimization algorithms. As the dimensionality of a problem increases, the search space volume grows exponentially, making it harder to locate the global optimum. Algorithms that perform well in low-dimensional spaces may struggle in high-dimensional environments. Therefore, global optimization algorithms are rigorously tested in high-dimensional search spaces to ensure robustness [21]. Scaling problems also pose difficulties for optimization algorithms. These problems involve significant variations in magnitude between the domain and the frequency of the hypersurface, complicating the search process [22]. For instance, Langerman functions are nonsymmetrical with randomly distributed local optima, making them particularly challenging to optimize. The quartic function introduces random noise (Gaussian or uniform), requiring algorithms to handle noisy data effectively. Algorithms that fail to perform well on noisy functions are likely to struggle in real-world applications where noise is prevalent. The benchmark functions used in this study are summarized in Tables 3 and 4, categorized by unimodal and multimodal functions, respectively. These functions are further classified based on characteristics such as continuity, separability, scalability, and differentiability. The GSM algorithm was compared against PSO, DE, BA, ABC, AHA, AHA-AO, CBO, ECBO, and SNS using the 22 benchmark functions listed in Tables 3 and 4. Each algorithm was executed 100 times, and the mean, standard deviation, and optimal values were recorded. The number of iterations and population size for each algorithm are detailed in Tables 3 and 4. All algorithms were implemented in MATLAB 24.2.0.2871072 (R2024b) Update 5 (The MathWorks, Inc., Natick, MA, USA), and the results are presented in Tables 5 and 6. The findings indicate that the GSM algorithm converges to optimal values with high precision, requiring fewer iterations and smaller population sizes compared to other algorithms. In Table 7, a \"+\" denotes cases where the GSM algorithm outperformed the other algorithms in terms of mean values, while a \"-\" indicates inferior performance. Overall, the GSM algorithm demonstrated superior performance on most benchmark functions compared to PSO, DE, BA, ABC, AHA, AHA-AO, CBO, ECBO, and SNS. The GSM algorithm demonstrates superior performance on unimodal, continuous, differentiable, nonseparable, and nonscalable functions \(F_{1}\), \(F_{5}\), and \(F_{6}\), outperforming the compared algorithms, including the most recent ones in the literature. Notably, for \(F_{1}\), the GSM algorithm consistently finds the global optimum with few iterations.