# Adaptive Multi-Level Search for Global Optimization:
## 5 Benchmark Problem
### Design of a Tension/Compression Spring
Other algorithms recently used to optimize this problem include co-evolutionary particle swarm optimization (CPSO) [22], differential evolution with dynamic stochastic selection (DEDS) [44], hybrid evolutionary algorithm with adaptive constraint-handling techniques (HEAA) [45], league championship algorithm (LCA) [46], water cycle algorithm (WCA) [24], and hybrid cuckoo search (HCS) [23]. A comparison of the optimal solutions by the aforementioned methods is given in Table 7, and the statistical results by ASIM, MAM, and other algorithms are shown in Table 8.  
In Table 7, the ASIM framework has the ability to find the optimal solution (0.0126652), which is the best available design compared to that which the other algorithms achieved. Although LCA [46] found a slighter better solution (0.01266523), the corresponding constraint \(g_{1}(x)\) was violated. Therefore, it was not a feasible solution. The same conclusion can be drawn for the results by DEDS [44] and HEAA [45]. Together with the statistical results shown in Table 8, it can be observed that the ASIM method is superior to other methods for the global solution in terms of the number of functio \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Number** & **Worst** & **Mean** & **Best** & **S.D.** & **NFEs** \\ \hline
1 & 0.012707419 & 0.01268076 & 0.012669372 & 1.53792 \(\times 10^{-5}\) & 4891 \\
2 & 0.015076822 & 0.013158868 & 0.012665512 & 0.001072278 & 5719 \\
3 & 0.012734131 & 0.012681909 & 0.012665469 & 2.94215 \(\times 10^{-5}\) & 5161 \\
4 & 0.013151181 & 0.012797596 & 0.012666127 & 0.000201587 & 5233 \\
5 & 0.012674725 & 0.012671127 & 0.012665294 & 3.80784 \(\times 10^{-6}\) & 4882 \\
6 & 0.012962267 & 0.012734387 & 0.012665259 & 0.000128041 & 5337 \\
7 & 0.012787169 & 0.012679022 & 0.012669651 & 1.17008 \(\times 10^{-5}\) & 4702 \\
8 & 0.012780362 & 0.01269988 & 0.012665634 & 4.68624 \(\times 10^{-5}\) & 5170 \\ \hline Average & 0.012669651 & 0.01266654 & 0.012665259 & 1.85492 \(\times 10^{-6}\) & 5141 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Statistical results for the tension/compression spring problem by adaptively integrated swarm intelligence-metamodelling (ASIM). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Number** & **Worst** & **Mean** & **Best** & **S.D.** & **NFEs** \\ \hline
1 & 0.032839737 & 0.015057587 & 0.0126692 & 0.004246608 & 8041 \\
2 & 0.046478999 & 0.01537479 & 0.012677425 & 0.005275199 & 8536 \\
3 & 0.070551755 & 0.015521846 & 0.012680762 & 0.009064574 & 7483 \\
4 & 0.053871312 & 0.016530777 & 0.012666692 & 0.00857695 & 7483 \\
5 & 0.030829567 & 0.014687079 & 0.012733211 & 0.003455907 & 7536 \\
6 & 0.017557055 & 0.014067046 & 0.012667273 & 0.001247161 & 8149 \\ \hline Average & 0.012733211 & 0.012682427 & 0.012666692 & \(2.55305\times 10^{-5}\) & 7871 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Statistical results for the tension/compression spring problem by MAM. accuracy throughout the optimization process. Obviously, the referenced methods used more than 10,000 calls to find the global optimum while ASIM finds the optimum with about half of those calls. Meanwhile, the ASIM could reduce the number of simulations by over 28% compared to MAM. As a general remark on the comparisons above, ASIM shows a very competitive performance over eight state-of-the-art optimization methods to find the global optimal solution in terms of the efficiency, the quality, and the robustness.