# Adaptive Multi-Level Search for Global Optimization:
## 5 Benchmark Problem
### Design of a Tension/Compression Spring  
This problem, first described by [PERSON] [43], has arisen from the wide applications of vibration resistant structures in civil engineering. The design objective is to minimize the weight of a tension/compression spring subject to constraints on the minimum deflection \(g_{1}\), shear stress \(g_{2}\), and surge frequency \(g_{3}\) and to limit on the outside diameter \(g_{4}\). As shown in Figure 5, the design variables include the wire diameter \(d\), the mean coil diameter \(D\), and the number of active coils \(N\). The mathematical description of this problem can be expressed as follows:  
\[ \begin{split} min\ f(N,\,D,\,d)=(N+2)\times Dd^{2}\\ s.t.\ g_{1}(\mathbf{x})=1-\frac{D^{3}N}{71,785d^{4}}\leq 0\\ g_{2}(\mathbf{x})=\frac{4D^{2}-Dd}{12,566(Dd^{3}-d^{4})}+\frac{1}{5 108d^{2}}-1\leq 0\\ g_{3}(\mathbf{x})=1-\frac{140.45d}{D^{2}N}\leq 0\\ g_{4}(\mathbf{x})=\frac{D+d}{1.5}-1\leq 0\\ \text{where }0.05\leq d\leq 1,\ 0.25\leq D\leq 1.3,\ 2\leq N \leq 15.\end{split}  \tag{19}\]  
The statistical results by MAM are in Table 5. From the first row to the sixth row, every row is the optimal results of 40 independent runs of MAM and the last line concludes the average results of the 6 parallel experiments, i.e., each experiment comprises 40 independent runs of MAM with randomly generated starting points. The best optimal design represented by \([d,D,N]\) is \([0.051656122,\,0.355902943,\,11.33791803]\) with the objective value of \(0.012666692\). Moreover, the fourth column \"Best\" in Table 5 indicates that MAM cannot achieve a converged robust solution and falls into the local optima when faced with multimodal function optimization. The optimal result ranges from \(0.01266\) (the best design in the fourth row) to \(0.070\) (the worst design in the third row). As a general deficiency of the trajectory-based algorithm, MAM could not find the known optimum \(0.0126652\) by balancing the efforts between exploration and exploitation. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Methods** & **Best** & **Average** & **Worst** & **S.D.** & **NFEs** \\ \hline GA-based [39] & 1.728226 & 1.792654 & 1.993408 & 0.074713 & 80000 \\ CPSO [22] & 1.728024 & 1.748831 & 1.782143 & 0.012926 & 200000 \\ ES-based [40] & 1.737300 & 1.813290 & 1.994651 & 0.070500 & 25,000 \\ CSS [41] & 1.724866 & 1.739654 & 1.759479 & 0.008064 & 4000 \\ CBO [42] & 1.724662 & 1.725707 & 1.725059 & 0.0002437 & 4000 \\ MAM & 1.724852 & 1.725563 & 1.739605 & 0.0031358 & 122 \\ ASIM & 1.724852 & 1.724852 & 1.724852 & 1.10E-07 & 565 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Statistical results from different optimization methods for the welded beam design problem. Figure 5: Schematic of the tension/compression spring. A more intuitive perspective for understanding the global search mechanism using the ASIM framework is represented in Table 6, which includes the optimal results obtained by 8 independent experiments, each of which is initialized with 5 particles. In Figure 6, the results show the objectives of initial designs and global optima for the tested 40 particles. Even the initial designs are remarkably different at the start of the optimization process due to the random nature of statistical tests; the developed ASIM has the capability to eventually find the converged global optimum. It is concluded that the ASIM algorithm can achieve a robust solution for random starting points, and it will not be trapped into local optima due to its multi-level search and model management strategies. Therefore, these 8 independent experiments could almost obtain the same global optimum. The best optimal design found by ASIM framework is \([0.051724501,0.357570887,11.23912608]\), with the objective value 0.012665259, which has a good agreement with the known optimum. Additionally, the global solutions from 8 independent experiments have been proven feasible by function evaluations.