# An Innovative Hybrid Approach Producing Trial Solutions for Global Optimization
## 3 Experiments
### Experimental Results
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**FUNCTION**} & **Proposed** & **Proposed** & **Proposed** \\  & **(Population = 200)** & **(Population = 150)** & **(Population = 100)** \\ \hline POTENTIAL3 & 4790 & 3655 & 2416 \\ \hline POTENTIAL5 & 7995 & 6710 & 4215 \\ \hline POTENTIAL6 & 8906 (76) & 8246 (70) & 4890 (33) \\ \hline POTENTIAL10 & 12,757 & 10,417 & 6598 (86) \\ \hline RASTRIGIN & 3672 (86) & 2525 (73) & 1728 (56) \\ \hline ROSENBROCK4 & 5642 & 4343 & 2995 \\ \hline ROSENBROCK8 & 7926 & 5953 & 4083 \\ \hline ROSENBROCK16 & 11,285 & 8694 & 5922 \\ \hline SCHWEFEL & 2208 & 1728 & 1125 \\ \hline SCHWEFEL221 & 2867 & 2192 & 1516 \\ \hline SCHWEFEL222 & 78,734 & 645,40 & 30,545 \\ \hline SHEKEL5 & 5322 & 3863 & 2407 \\ \hline SHEKEL7 & 5503 & 3568 & 2556 \\ \hline SHEKEL10 & 4784 & 3277 & 2374 \\ \hline SINU4 & 3446 & 2561 & 1858 \\ \hline SINU8 & 4371 & 3292 & 2258 \\ \hline SINU16 & 6450 & 5162 & 3386 \\ \hline TEST2N4 & 3569 & 2597 & 1988 (86) \\ \hline TEST2N5 & 3877 (93) & 3130 (96) & 1915 (80) \\ \hline TEST2N7 & 4285 (50) & 3325 (36) & 2089 (33) \\ \hline TEST30N3 & 4215 & 2756 & 1870 \\ \hline TEST30N4 & 5013 & 3768 & 2115 \\ \hline SUM & 361,454 & 278,124 & 174,545 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: _Cont._proposed methods, with 5255 calls at 10 dimensions and 26,684 at 100. GENETIC displays a variable pattern, with 3131 calls at 10 dimensions and 29,004 at 100. GWO and PSO follow a similar trend, with a gradual increase in calls as the dimension increases. Additionally, Figure 3(b) illustrates the execution time for each algorithm across different dimensions, measured in seconds. As expected, increasing the dimension leads to a significant rise in execution time for all algorithms, highlighting the increased computational load. For example, PROPOSED (KMEANS) starts with 0.652 s at 10 dimensions and reaches 23.54 s at 100. PROPOSED begins with 0.236 s at 10 dimensions and climbs to 32.784 s at 100, marking an impressive 139-fold increase. The fastest algorithm, IPSO, takes only 0.034 s at 10 dimensions and 2.106 s at 100, demonstrating its excellent performance and scalability. On the other hand, WOA records the worst performance, with 75.319 s at 100 dimensions compared to 0.257 s at 10 dimensions. DE also struggles with its execution time, increasing from 0.16 s at 10 dimensions to 47.68 s at 100. The increase in execution time across dimensions follows a nonlinear trend for most algorithms. For example, PROPOSED takes 0.587 s at 20 dimensions (more than double the 0.236 s at 10 dimensions) but reaches 14.847 s at 80 dimensions. Similarly, PSO starts at 0.128 s at 10 dimensions and reaches 20.361 s at 100 dimensions. The data reveal significant differences in the scalability and efficiency of the algorithms. IPSO stands out as the most efficient algorithm, with the lowest number of calls and the shortest execution time. PROPOSED (KMEANS) also achieves satisfactory results, maintaining a balance between calls and execution time. On the other hand, WOA and GWO exhibit the worst performance, with significantly more calls and substantial increases in execution time as the number of dimensions increases. The results confirm that the complexity of the optimization problems increases sharply with the dimension. Most algorithms struggle to maintain their efficiency at higher dimensions, highlighting the importance of selecting the appropriate algorithm according to the problem's requirements and the available computational resources. In each iteration of the algorithm, a trial point is calculated through vector operations, similar to the process of optimization that uses differential evolution. The main difference between the proposed method and differential evolution lies in the fact that the samples  
Figure 4: Different variations of the ELP function of the proposed method and other methods. used for calculating the trial point are selected from nearby regions of the initial distribution, rather than being chosen randomly.