# Use RBF as a Sampling Method in Multistart Global Optimization Method
## 3 Experiments
### Experimental Results  
The proposed sampling method was tested against the uniform sampling, for the Multistart global optimization technique. The uniform distribution used to sample points is defined as:  
\[x_{i}=a_{i}+r\times(b_{i}-a_{i}),\ i=1\ldots n \tag{10}\]  
with \(r\in[0,1]\) a random number. The parameters for the experiments are listed in Table 1. All the experiments were executed 30 times with different random numbers each time. In all cases, the stopping rule of Section 2.2 was incorporated. The random number function used was the drand480 function of the C programming language. The used software was implemented using the OPTIMUS global optimization environment freely available from [[https://github.com/itsoulos/OPTIMUS](https://github.com/itsoulos/OPTIMUS)]([https://github.com/itsoulos/OPTIMUS](https://github.com/itsoulos/OPTIMUS)) (accessed on 27 November 2022). All the experiments were conducted on an AMD Ryzen 5950X equipped with 128 GB of RAM. The operating system used was Debian Linux and all the programs are compiled using the GNU C++ compiler. The experimental results for used test functions are listed in Tables 2-4. The number in the cells denotes the average function calls for the 30 independent runs. The fraction in parentheses stands for the fraction of runs where the global optimum was found. If this number is missing then the global minimum was discovered in every independent run (100% success). At the end of each table, an additional line named total has been added,representing the total number of function calls and, in parentheses, the average success rate in finding the total minimum. From the experimental results, it follows in principle that the use of neural networks significantly reduces the required number of function calls needed to find the total minimum. This reduction is proportional to the objective function and can reach up to 60% of the original number of function calls. In addition, the usage of the ISAMPLES parameter significantly increases the reliability of the new sampling method. For example, in the case where N = 20 there is an increase in the average success rate for finding the total minimum from 90% to approximately 96%. However, improving the reliability of the method does not imply an increase in the number of function calls. For example, for N = 20 the calls are in the interval \([75,000\ldots 90,000]\) without showing any clear increase. However, the use of the new sampling technique requires significantly more computing time than the uniform distribution because of the need to train the RBF networks and also because of the classification that precedes taking the final samples. This difference is demonstrated in Figure 6. In this we see the significant difference in running time for the SINU problem with a different number of dimensions each time. \begin{table}
\begin{tabular}{c c} \hline \hline
**PARAMETER** & **VALUE** \\ \hline ITER\({}_{\mbox{MAX}}\) & 100 \\ \hline \(k\) & 10 \\ \hline \(N\) & 20,50 \\ \hline FR & \(10\times N\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters for the experiments. \begin{table}
\begin{tabular}{c c c} \hline \hline
**FUNCTION** & **N = 20** & **N = 50** \\ \hline BF1 & 3004 & 5975 \\ \hline BF2 & 2828 & 5826 \\ \hline BRANIN & 2409 & 5415 \\ \hline CAMEL & 2661 & 5599 \\ \hline CIGAR & 5588 & 8410 \\ \hline CM4 & 3551 (0.87) & 6431 (0.80) \\ \hline DISCUS & 2817 & 5965 \\ \hline EASOM & 2204 & 5202 \\ \hline EXP4 & 2769 & 5772 \\ \hline EXP16 & 2836 & 5837 \\ \hline EXP64 & 2912 & 5914 \\ \hline GRIEWANK2 & 3938 (0.40) & 6572 (0.30) \\ \hline GRIEWANK10 & 4536 (0.97) & 7520 \\ \hline POTENTIAL3 & 3121 & 6120 \\ \hline POTENTIAL5 & 4363 & 7320 \\ \hline HANSEN & 5344 (0.93) & 9536 (0.90) \\ \hline HARTMAN3 & 2618 & 5608 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Experimental results for the Multistart method, using uniform distribution for the samples as defined in Equation (10).