## 3 Optimization Scheme
### The First-Level Approximation Problems by BMA  
The branched multipoint approximation (BMA) method has proven to be one of the most effective approximation techniques for complicated structure optimization problems [26] and is applied to approximate functions of the primal problem in Equation (1). The function \(g_{j}(\mathbf{X},\mathbf{\alpha})\) is approximated as \(g_{j}^{(p)}(\mathbf{X},\mathbf{\alpha})\) in Equations (2)-(6). In the \(p\)-th stage, \(g_{j}^{(p)}(\mathbf{X},\mathbf{\alpha})\) is the \(j\)-th approximate constraint function at the \(p\)-th stage. \[g_{j}^{(p)}(\mathbf{X},\mathbf{\alpha})=\sum_{t=p-(H-1)}^{H}\Bigg{\{}g_{j}(\mathbf{X}_{t}, \mathbf{\alpha}_{t})+\sum_{i=1}^{n}\tilde{g}_{j,i,t}(\mathbf{X},\mathbf{\alpha})\Bigg{\}}h_ {t}(\mathbf{X},\mathbf{\alpha}) \tag{2}\]  
\[\tilde{g}_{j,i,t}(\mathbf{X},\mathbf{\alpha})=\left\{ \begin{array}{l}\frac{1}{r_{o,t} }\frac{\partial g_{j}(\mathbf{X}_{t},\mathbf{\alpha}_{t})}{\partial x_{i}}x_{it}^{1-r _{o,t}}\bigg{(}x_{i}^{r_{o,t}}-x_{it}^{r_{o,t}}\bigg{)},\;if\;\alpha_{i}=1\\ \\ \frac{1}{r_{m,t}}\frac{\partial g_{j}(\mathbf{X}_{t},\mathbf{\alpha}_{t})}{\partial x_ {i}}\Big{(}1-e^{-r_{m,t}(x_{i}-x_{it})}\Big{)},\;if\;\alpha_{i}=0\end{array} \right. \tag{3}\]  
\[h_{t}(\mathbf{X},\mathbf{\alpha})\ =\ \frac{\overline{h}_{t}(\mathbf{X},\mathbf{\alpha})}{\sum _{i=1}^{H}\overline{h}_{t}(\mathbf{X},\mathbf{\alpha})}\;t=1,\cdots,H \tag{4}\]  
\[\overline{h}_{t}(\mathbf{X},\mathbf{\alpha})=\prod_{s=1,s\
eq l}^{H}(\mathbf{X}-\mathbf{X}_{s })^{T}(\mathbf{X}-\mathbf{X}_{s}) \tag{5}\]  
\[\left\{ \begin{array}{l}\min\sqrt{\left\{\sum_{z=1}^{H}\{g_{j}(\mathbf{X}_{z}, \mathbf{\alpha}_{z})-g_{j}(\mathbf{X}_{t},\mathbf{\alpha}_{t})-\tilde{g}_{j,i,t}(\mathbf{X}_{ t},\mathbf{\alpha}_{t})\}\right\}^{2}}\\ s.t.\ \ -5\leq r_{o,t}\leq 5,\ \ -5\leq r_{m,t}\leq 5,\ \ \ t=1,\cdots,H-1\end{array} \right. \tag{6}\]  
where \(x_{t}\) is the \(t\)-th known point (from structural analyses), \(H\) is the number of counted points in the approximate function, and \(H=\min\{p,H_{\max}\}\). When the number of known points is larger than \(H_{\max}\) (in this work, \(H_{\max}=5\)), only the last \(H_{\max}\) points are included in the function. \(\overline{\mathbf{X}}\) is the convex domain that limits the bounds of \(x\); \(h_{t}(\mathbf{X},\mathbf{\alpha})\) is a weighting function, which is defined in Equations (4) and (5). \(r_{o,t}\) and \(r_{m,t}\) are the adaptive parameters controlling the nonlinearity of \(g_{j}^{(p)}(\mathbf{X},\mathbf{\alpha})\), which are determined by solving the least-squares parameter estimation in Equation (6). Here, \(g_{j}(\mathbf{X}_{z},\mathbf{\alpha}_{z})\) is the \(j\)-th constraint value of the t-th known point. For every value of t, a parameter \(r_{t}(r_{o,t}\) or \(r_{m,t})\) will be obtained from Equation (6). To avoid large fluctuations in the approximation formula, the value range of \(r_{t}\) (\(r_{o,t}\) or \(r_{m,t}\)) should be limited to within \((-5,\,5)\). If t equals 1, then \(r_{o,t}=-1\) and \(r_{m,t}=3.5\). It can be seen that the construction of the approximate function only requires the partial derivatives of the continuous variables, and discrete variables only act the role of branching, deciding \(\alpha_{1}\) to equate to which value, i.e., 0 or 1. The form of the BMA function is similar to the Hermite interpolation function, and in the known points, the value of the approximate function and their derivatives are consistent with the values in the primal function. This approximate function has been proven to be efficient without singularity even if the continuous variables are close to zero. The approximation accuracy of BMA functions was discussed in [27; 28]. To solve the problem in Equation (1), which is always implicit and nonlinear, a first-level approximation problem is first constructed to transform it into an explicit problem. Based on the above BMA method, the first-level approximation problem in the \(p\)-th stage is:  
\[\left\{