# An Innovative Hybrid Approach Producing Trial Solutions for Global Optimization
## 3 Experiments
### Experimental Results  
Different seeds were used in a random number generator to ensure the reliability of the experimental results, with the experiments being repeated 30 times. This process of repetition aims to minimize the likelihood of random errors and to enhance the validity of the results. The experiments were conducted on a system with an AMD Ryzen 5950X processor and 128 GB of RAM, operating in a Linux-Debian environment. Additionally, the open-source optimizer \"GlobalOptimus version 1.0\" was used, which is a fully developed optimizer and is available for distribution via the following link: [[http://www.github.com/itsoulos/GlobalOptimus](http://www.github.com/itsoulos/GlobalOptimus)]([http://www.github.com/itsoulos/GlobalOptimus](http://www.github.com/itsoulos/GlobalOptimus)) (accessed on 7 November 2024). This optimization software can be installed on any operating system (Linux, Windows, FreeBsd) when the Qt library is available. The software includes a variety of optimization methods as well as some scripts to automate the optimization process under Windows or Linux. Also, the software provides a graphical user interface under the name Xoptimus and an installation file without dependencies for the Windows system is also included. In Table 3, the average function calls for each method and every objective function are presented. The following notations are used in the experimental tables:  
1. The column FUNCTION represents the name of the objective problem used. 2. The column GENETIC represents the application of the genetic algorithm to the objective problem [12]. 3. The PSO column represents the implementation of the classical PSO algorithm as suggested by the literature [14]. 4. The column IPSO stands for the application of the Improved PSO algorithm, as suggested by [PERSON] and [PERSON] [34], to the objective problem. 5. The column DE represents the average function calls for the differential evolution optimization technique [43]. 6. The column GWO represents the average function calls for the Gray Wolf optimization technique [35]. 7. The column WOA represents the average function calls for the Whale Optimization technique [18]. 8. The column MEWOA stands for the application of the Improved WOA algorithm, as suggested by [PERSON] et al. [44], to the objective problem. 9. The total sum of the calls for each method is listed at the end of the table. 10. The column PROPOSED stands for the application of the current method. 11. The column KMEANS denotes the incorporation of KMEANS sampling into the current method. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Value** & **Explanation** \\ \hline \(N\) & 200 & Number of samples for all methods \\ \hline \(N_{k}\) & 200 & Maximum number of iterations for all methods \\ \hline \(SR\) & \(\left|f_{\min}^{(k)}-f_{\min}^{(k-1)}\right|\) & Best fitness: stopping rule for all methods \\ \hline \(N_{t}\) & 12 & Similarity max count for all methods \\ \hline \(LS\) & 0.05 (5\%) & Local search rate for all methods \\ \hline \(F\) & 0.8 & Differential weight for differential evolution \\ \hline \(CR\) & 0.9 & Crossover probability for differential evolution \\ \hline \(C_{1},C_{2}\) & 0.5 & Parameters of PSO \\ \hline \(G_{c}\) & 0.1 (10\%) & Crossover rate for genetic algorithm \\ \hline \(G_{m}\) & 0.05 (5\%) & Mutation rate for genetic algorithm \\ \hline \(M_{k}\) & 10 & Number of samples of sub-population for MEWOA \\ \hline \(M_{r}\) & 1 & Number of iterations of exchange information for MEWOA \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Settings of parameters used in the optimization methods.