## 2 Materials and Methods
### Application of Test Functions  
In this study, we conduct a benchmarking analysis of multiple population-based metaheuristic algorithms. The algorithms under scrutiny include the QSA, ABC, GA, PSO, SA, CS, and OPSO. The evaluation is conducted using two well-known continuous test functions: the HyperSphere (Sphere) function and the Levy function. These test functions are commonly used to evaluate the performance of optimization algorithms in both unimodal and multimodal search spaces. The process of benchmarking is conducted through the utilisation of Python programming. The Python and Matlab code for QSA and other algorithms can be downloaded from the GitHub repository at [[https://github.com/OQSA/OQSA](https://github.com/OQSA/OQSA)]([https://github.com/OQSA/OQSA](https://github.com/OQSA/OQSA)). Definitions of Test Functions  
We applied the HyperSphere and the Levy function for benchmarking. The HyperSphere function is representative of a unimodal and convex optimization problem. It provides a simple test for the convergence speed and numerical accuracy of optimization algorithms in a smooth landscape. In contrast, the Levy function exhibits a complex, multimodal search space with many local optima. It is used to evaluate an algorithm's exploration capabilities and its robustness in avoiding premature convergence to local minima. **HyperSphere (Sphere) Function**: The HyperSphere function, also known as the Sphere function, is a fundamental test function used in optimization. It is mathematically defined as follows:  
\[f_{\text{Sphere}}(x)=\sum_{i=1}^{n}x_{i}^{2}, \tag{23}\]  
where  
\[x=(x_{1},x_{2},\ldots,x_{n})\in\mathbb{R}^{n}. \tag{24}\]  
**Global Optimum:** The function achieves its global minimum at  
\[f(0)=0,\quad\text{when}\quad x=0. \tag{25}\]  
This implies that the best possible solution occurs when all variables are set to zero. **Unimodal Nature:** The function is unimodal. This property, which is defined shortly, ensures that optimization algorithms do not become trapped in local minima. Instead they reach a single global minimum. For this reason, the function has been proven to be an excellent benchmark for global optimization techniques. **Convex and Smooth Search Space:** It has been demonstrated that the function is convex, which guarantees that any local search method eventually converges to the global minimum. Additionally, its smooth gradient makes it highly suitable for gradient-based optimization algorithms. **Scalability:** Since the function is defined in an n-dimensional space, it is widely used to benchmark optimization algorithms for different problem sizes. The computational complexity of evaluating the function grows linearly with the number of dimensions, making it a scalable test function. **Applications:** The HyperSphere function is widely utilized in optimization research for benchmarking **metaheuristic optimization** algorithms such as GA, PSO, and Differential Evolution (DE), evaluating the performance of algorithms on **continuous, differentiable, and convex landscapes**. Analyzing the behavior of optimization methods in **simple, gradient-friendly** search spaces before applying them to more complex functions. **Conclusion:** The HyperSphere function serves as a fundamental benchmark in optimization studies. Its convex and smooth nature makes it an ideal test function for evaluating the performance of different optimization methods before applying them to real-world, complex, and multi-modal problems. **The Levy Function:** The Levy function is utilized as a benchmark test function in the present comparative study of metaheuristic algorithms. This function is widely recognized for its complex, multi-modal landscape, which poses a significant challenge for global optimization methods.