## 3 Numerical Study of Developed Algorithms
### The Sutton-Chen Problem  
We consider finding the low-energy Sutton-Chen clusters task [37; 38; 39; 40] for extremely large dimensions. The Sutton-Chen potential is often used in computations of the nanocluster properties of metals such as silver, rhodium, nickel, copper, gold, platinum, and others [38; 39; 40]. The following global optimization problem is considered:  
\[ \begin{split} f(x)=\epsilon\Sigma\left[\frac{1}{2}\sum\limits_{j \
eq i}\left(\frac{\hat{x}}{\hat{x}_{ij}}\right)^{q}-c\sqrt{\sum\limits_{j \
eq i}\left(\frac{\hat{x}}{\hat{x}_{ij}}\right)^{m}}\right]\to\min,\\ r_{ij}=\sqrt{\left(x_{3(i-1)+1}-x_{3(j-1)+1}\right)^{2}+\left(x_{3( i-1)+2}-x_{3(j-1)+2}\right)^{2}+\left(x_{3(i-1)+3}-x_{3(j-1)+3}\right)^{2}}. \end{split}  \tag{6}\]  
Here, \(\epsilon=1.0,\ c=144.41,\ a=1.0,\ q=12.0,\ m=6.0\) are special parameters. To investigate the problems of the optimization of atomic-molecular cluster potentials, we implemented three-phase computational technology, which combines several methods from a collection of basic at each phase [41]. Phase 1 is concentrated on the random approximation of a neighborhood of a record point and established on a collection of algorithms that generates starting approximations: \"level\" (LG), \"averaged\" (AG), \"gradient\" (GG), \"level-averaged\" (LAG), \"level-gradient\" (LGG), \"averaged-gradient\" (AGG), \"level-averaged-gradient\" (LAGG). At the phase 2, the problem of descent of the approximations produced at the phase 1 [42] is determined by applying one of the \"starter algorithms\" that can find low-energy configurations of atoms. We implemented six variants of \"starter algorithms\": three bioinspired algorithms described in the paper (FP, TL, FA), the variant of the method proposed by [PERSON] [43; 44], the \"raider method\" [45], and the modification of the multivariate dichotomy method [46; 47]. In all these algorithms we applied the modification of trust region method for local search (phase 3). Nowadays, solutions are presented only for Sutton-Chen configurations up to 80 atoms in the Cambridge Energy Landscape Database [29] and other public sources. We made a performance investigation of developed techniques for clusters of 3-80 atoms and compared the solutions obtained with the best of the known ones [48]. The solutions obtained during numerical study coincided with the known results represented in [29]. We investigated very large problems for configurations of atoms from 81 to 100. For each algorithm-generator and each starter method, we performed computational experiments. We used an excess of the time limit (24 h) as a stopping criterion. Table 3 presents the results of the numerical testing of computational technology for the task in which the cluster consists of 85 atoms. As can be noticed from Table 3, there is a leading computational technology with a combination of the LAGG as an initial approximation generator and the raider algorithm as a starter, among all the investigated variants. Table 4 presents the best-found values obtained. It was not possible to compare our results with the computations of other researchers due to the absence of such information. Figure 5 demonstrates the dependency of the best-found values on the dimension. We plot the number of atoms along the horizontal axis, the found values along the vertical. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**N** & **Value** & **N** & **Value** & **N** & **Value** \\ \hline
81 & \(-\)79,382.0486 & 86 & \(-\)84,502.6321 & 91 & \(-\)89,817.3875 & 96 & \(-\)95,046.4031 \\
82 & \(-\)80,478.7411 & 87 & \(-\)85,627.6685 & 92 & \(-\)90,787.2961 & 97 & \(-\)96,121.8664 \\
83 & \(-\)81,403.5270 & 88 & \(-\)86,607.6112 & 93 & \(-\)91,827.6375 & 98 & \(-\)97,165.4582 \\
84 & \(-\)82,468.2286 & 89 & \(-\)87,689.9154 & 94 & \(-\)92,906.8617 & 99 & \(-\)98,222.4219 \\
85 & \(-\)83,552.3954 & 90 & \(-\)88,776.5998 & 95 & \(-\)93,916.8247 & 100 & \(-\)99,273.6113 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The best-found values (number of atoms: from 81 to 100).