# A Hybrid Adaptive Particle Swarm Optimization Algorithm for Enhanced Performance
## 4 Experiments  
The CEC benchmarking function provides a standardized framework for assessing the efficiency of optimization algorithms across various scales [32]. It incorporates complex mathematical formulations that are both intricate and diverse, playing a crucial role in effectively measuring and evaluating the optimization performance of these algorithms. Through comparative analyses of different optimization algorithms, researchers can refine tuning processes to achieve better optimization results. To thoroughly assess the capabilities of the proposed algorithms, benchmark functions with diverse characteristics in Table 2 are utilized for testing [32]. These functions encompass unimodal, multimodal, and composite benchmarking types, and are very suitable for verifying the key indexes of the algorithm such as the convergence speed, the local search accuracy, the global exploration ability, and the high-dimensional adaptability. This study demonstrates the comprehensive performance of the proposed algorithm by randomly selecting benchmark functions and comparing it with the standard PSO algorithm [26], as well as several particle swarm enhancement algorithms (e.g., NIPAO [27], IMPSO [28], CLPSO [2], IFEHPSO [29]) and other metaheuristic algorithms (e.g., GWO [33], WOA [34], SSA [35], BOA [33]). To ensure the reliability of the simulation test results for each algorithm and to enable a fair comparison of their performances, the population size for the particle swarm and the number of iterations were standardized to 30 and 200, respectively, with each benchmark function executed independently 30 times. To facilitate clear observation and comparison of the test results from the benchmark functions, the average (\(V_{avg}\)), standard deviation (\(V_{std}\)), and optimal value (\(V_{opt}\)) of the corresponding results are employed as evaluation metrics, with \(V_{opt}\) highlighted as the primary performance index. The experiments were conducted on a server equipped with a CPU (Intel Core i7, 3.20 GHz, 12 cores, 24 threads) and 32 GiB of DDR4 RAM (3600 MHz). The operating system was Windows 10, the integrated development environment was Spyder 5.4, and the experimental programs were written in Python 3.12. \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Function Name** & **Formula** & **Range** & **Type** \\ \hline Sphere & \(F_{1}(x)=\sum_{i=1}^{n}x_{i}^{2}\) & \([-100,100]\) & S \\ Schwefel2.22 & \(F_{2(x)}=\sum_{i=1}^{n}|x_{i}|+(\Gamma_{i=1}^{n}|x_{i}|)\) & \([-10,10]\) & S \\ [PERSON] & \(F_{3}(x)=\sum_{i=1}^{n}\left(\sum_{j=1}^{n}x_{j}\right)^{2}\) & \([-100,100]\) & S \\ [PERSON] & \(F_{4}(x)=\max_{i}\left\{|x_{i}|,\,1\leq i\leq n\right\}\) & \([-100,100]\) & [PERSON] \\ [PERSON] & \(F_{5}(x)=\sum_{i=1}^{n}\left(100\left(x_{i+1}+x_{i}^{2}\right)+(x_{i}-1)^{2}\right)\) & \([-30,30]\) & [PERSON] \\ [PERSON] & \(F_{6}(x)=\sum_{i=1}^{n}(x_{i}+0.5)^{2}\) & \([-100,100]\) & S \\ Quartic & \(F_{7}(x)=\sum_{i=1}^{n}i*x_{i}^{4}+rand[0,1]\) & \([-1.28,1.28]\) & M \\ Rastrigrin & \(F_{8}(x)=\sum_{i=1}^{n}|x_{i}^{2}-10 cos(2\pi x_{i})+10]\) & \([-500,500]\) & M \\ [PERSON] & \(F_{9}(x)=\frac{290}{100}\sum_{i=1}^{n}x_{i}^{2}-\Gamma_{i=1}^{n}\cos\left( \frac{x_{i}}{\sqrt{i}}\right)+1\) & \([0,1]\) & M \\ Penalized2 & \(F_{10}(x)=\sin^{2}(3\pi x_{1})+\sum_{i=1}^{n}(x_{i}-1)^{2}[1+\sin^{2}\left(3 \pi x_{i}+1\right)]+(x_{n}-1)^{2}[1+\sin^{2}\left(2\pi x_{n}\right)]+\) & \([-50,50]\) & M \\ Penalized1 & \(F_{11}(x)=\frac{\pi}{\pi}\left\{10\sin\left(\pi y_{1}\right)+\sum_{i=1}^{n} \left(y_{i}-1\right)^{2}\left(1+10\sin^{2}\left(\pi y_{i+1}\right)\right)+\left( y_{n}-1\right)^{2}\right\}\) & \(+\) & \([-50,50]\) & M \\  & \(\sum_{i=1}^{n}u(x_{i},10,100,4)\) & \([-1,1]\) & B \\ Sum of Powers & \(F_{12}(x)=\sum_{i=1}^{d}|x_{i}|^{i+1}\) & \([-1,1]\) & T \\ Booth & \(F_{13}(x)=(x_{1}+2x_{2}-7)^{2}+(2x_{1}+x_{2}-5)^{2}\) & \([-10,10]\) & T \\ [PERSON] & \(F_{14}(x)=(x_{1}-1)^{2}+\sum_{i=1}^{d}2(x_{i}^{2}-x_{i-1})^{2}\), & \([-10,10]\) & V \\ \hline \hline \end{tabular}
\end{table}