# Adaptive Multi-Level Search for Global Optimization:
## 5 Benchmark Problem
### Welded Beam  
The design optimization of a welded beam in Figure 4 is a complex and challenging problem in nature with many variables and constraints. Usually, conventional optimization methods fail to find global optimal solutions. Hence, the welded beam design problem is often used to evaluate the performance of optimization methods. To determine the best set of design variables for minimizing the total fabrication cost of the structure, the minimum cost optimization is performed considering shear stress \((\tau)\), bending stress \((\sigma)\), buckling load \((p_{c})\), and end deflection \(\delta\) constraints. The design variables comprise the thickness of the weld \((x_{1})\), the length of the welded joint \((x_{2})\), the width of the beam \((x_{3})\), and the thickness of the beam \((x_{4})\), and the mathematical formulation of this problem can be expressed as follows:  
\[ \begin{array}{ll}min&f(\mathbf{x})=1.10471x_{1}^{2}x_{2}+0.04811x_{3}x_{4}(14+x _{2})\\ s.t.&g_{1}(\mathbf{x})=\tau(\mathbf{x})-\tau_{max}\leq 0\\ &g_{2}(\mathbf{x})=\sigma(\mathbf{x})-\sigma_{max}\leq 0\\ &g_{3}(\mathbf{x})=x_{1}-x_{4}\leq 0\\ &g_{4}(\mathbf{x})=[0.10471x_{1}^{2}+0.04811x_{3}x_{4}(14+x_{2})]-5\leq 0\\ &g_{5}(\mathbf{x})=0.125-x_{1}\leq 0\\ &g_{6}(\mathbf{x})=\delta(\mathbf{x})-\delta_{max}\leq 0\\ &g_{7}(\mathbf{x})=p-p_{c}(\mathbf{x})\leq 0\\ \end{array} \]  
_where_  
\[ \begin{array}{ll}P=6000\,lb,\ L=14\,in,\ E=30\times 10^{6}\,psi,\ G=12\times 1 0^{6}\,psi,\\ \tau_{max}=13,600\,psi,\ \sigma_{max}=30,000\,psi,\ \delta_{max}=0.25\,in \\ &\tau^{\prime}=\frac{P}{\sqrt{2}x_{1}x_{2}},\tau^{\prime\prime}=\frac{ MR}{J},M=P\Big{(}L+\frac{x_{2}}{2}\Big{)}\\ & R=\sqrt{\frac{x_{2}^{2}}{4}+\left(\frac{x_{1}+x_{3}}{2} \right)^{2}}\\ &\tau(x)=\sqrt{\left(\tau^{\prime}\right)^{2}+2\tau^{\prime}\tau^{-\frac{x_{ 2}^{2}}{2R}+\left(\tau^{\prime\prime}\right)^{2}}}\\ & J=2\Bigg{\{}\sqrt{2}x_{1}x_{2}\Bigg{[}\frac{x_{2}^{2}}{12}+ \left(\frac{x_{1}+x_{3}}{2}\right)^{2}\Bigg{]}\Bigg{\}}\\ &\sigma(x)=\frac{6 PL}{x_{4}x_{3}^{2}},\delta(x)=\frac{4 PL^{3}}{Ex_{3}^{3}x_{4}} \\ & p_{c}(x)=\frac{4.013\sqrt{E\left(\frac{x_{2}^{2}x_{4}^{6}}{36} \right)}}{L^{2}}\Bigg{(}1-\frac{x_{3}}{2L}\sqrt{\frac{E}{4G}}\Bigg{)}\\ & 0.1\leq x_{1}\leq 2,\ 0.1\leq x_{2}\leq 10,\ 0.1\leq x_{3}\leq 10,\ 0.1\leq x_{4}\leq 2 \end{array} \]  
To solve the aforementioned problem, the GA-based method [39], co-evolutionary PSO method (CPSO) [22], ES-based method [40], charged system search (CSS) [41], and colliding bodies optimization (CBO) [42] were used to find the optimal solution. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multicolumn{3}{c}{**MAM Parameters**} & \multicolumn{3}{c}{**PSO Parameters**} \\ \cline{2-7}  & **MI a** & **NOP b** & **SIR c** & **SMR d** & **\(\mathbf{\omega}\) e** & **a f** & \(\mathbf{\beta}\) f \\ \hline \multirow{2}{*}{MAM} & 30 & \(n+5\) & 0.25 & 0.1 & & N.A. & \\  & 30 & \(n+5\) & 0.25 & 0.1 & 0.7298 & 1.49618 & 1.49618 \\ \hline \hline \end{tabular}  
* The maximum number of iteration. b The number of required sampling points. c The size of the initial trust region. d The minimum size of the trust region. e The initial weight in PSO. f The acceleration coefficients in PSO.  
\end{table} 
Table 2: Default parameters for MAM and ASIM. In Table 3, the optimized design variables and cost obtained by MAM and ASIM have been compared with those obtained in literature. The best solutions (1.724852) by MAM and ASIM are more competitive than those obtained by other methods. Although [PERSON] [42] claimed that 1.724663 was the better cost, the solution actually violated the \(g_{1}\) constraint and it was an infeasible solution. Based on the statistical results in Table 4, it is concluded that the ASIM technique is very robust and efficient because the standard deviation of different runs of simulations is almost 0 (\(1.1\times 10^{-7}\)) and the number of function analysis (NFEs) is remarkably smaller (565) than that called by other methods except MAM.