## 4 The Proposed Algorithm
### Iteration Framework  
Theoretically, the EBDO problems in Equation (9) can be solved by existing methods [34]. However, the resulting computational burden will be extremely heavy. To address this issue, a novel iteration framework is developed, in which the uncertainty analysis and design optimization alternate until convergence. In the \(k\)-th iteration, each optimization problem in Equation (9) requires the performance of an uncertainty analysis at the previous design point:  
\[Bel\left(f(\mathbf{Z})\leq v_{j}^{(k-1)}\right)\text{, }\mathbf{Z}=(\mathbf{X},\mathbf{P})\text{, }j=1,2,\ldots,n_{T} \tag{10}\]  
This mainly consists of two steps, illustrated by the example in Figure 1. Step 1 is to search for the most probable focal element (MPFE) along the limit-state boundary of \(f(\mathbf{Z})\leq v_{j}^{(k-1)}\). The MPFE [35] is similar to the most probable point (MPP) in probability theory, which is the point with the most probability density on the limit-state boundary. Compared to other points on the boundary, the minimal error of reliability analysis can be achieved by establishing the linear approximation for the performance function at the MPP [36]. Similarly, the MPFE contains the maximal BPA among the focal elements that are crossed by the limit-state boundary. The searching process of MPFE is formulated as:  
\[\max_{D_{k}}m(D_{k}) \tag{11}\] \[\text{s.t. }f(\mathbf{Z})=v_{j}^{(k-1)}\text{ }\Bigg{\}}j=1,2, \ldots,n_{T}\]  
where \(m(D_{k})\) represents the BPA of the focal element where the \(\mathbf{Z}\) point is located. Note that there is a difference between \(v_{j}^{(k-1)}\), \(j=1,2,\ldots,n_{T}\) at each iteration step due to the minor difference of \(Bel_{j}^{T}\). Consequently, different MPFEs may be obtained for Equation (11). However, the difference between the MPFEs is minor relative to the entire design domain. To ensure efficiency, the unique MPFE is investigated at each iteration. Equation (11) can be rewritten as:  
\[\max_{D_{k}}m(D_{k}) \tag{12}\] \[\text{s.t. }f(\mathbf{Z})=v^{(k-1)}\]  
where \(v^{(k-1)}\) represents the performance threshold that has not yet converged. Step 2 is to establish linear approximation for the performance function at the central point \( \begin{pmatrix}\text{-}M\\ \mathbf{Z}\end{pmatrix} \) of MPFE:  
\[L^{(k)}(\mathbf{Z})=f\Big{(}\mathbf{Z}^{M(k)}\Big{)}+\Big{(}\mathbf{Z}-\mathbf{Z}^{M(k)} \Big{)}^{\text{T}}\cdot\
abla f\Big{(}\mathbf{Z}^{M(k)}\Big{)} \tag{13}\]  
The \(L\)-function is used to replace the \(f\)-function to calculate \(Bel\), and thereby the optimization processes in Equation (7) no longer requires the calculation of any performance function. The efficient calculation of \(Bel\) has been achieved in the iterative process, but the overall process of EBDO still requires dozens or even hundreds of \(Bel\) evaluations due to the nested optimization. To eliminate the nested optimization, a decoupling strategy is proposed similar to that in the probabilistic method [37]. At each iteration step, the reliability constraint is transformed into a deterministic constraint by constructing the shifting vector of \(S_{j}^{(k)}\); and then a deterministic optimization is updated and solved to obtain the current solution. In the \(k\)-th iteration, the deterministic optimization can be written as:  
\[ \begin{array}{l}\min v\\ \text{s.t. }f\left(\boldsymbol{d},\boldsymbol{Z}-\boldsymbol{S}_{j}^{(k)} \right)\leq v\\ \boldsymbol{d}^{l}\leq\boldsymbol{d}\leq\boldsymbol{d}^{u},\ \overline{\boldsymbol{X}}^{l}\leq \boldsymbol{X}\leq\overline{\boldsymbol{X}}^{u}\end{array} \right\}j=1,2,\ldots,n_ {T} \tag{14}\]  
The shifting vector determines the deviation between the original reliability boundary and the deterministic boundary at the \(k\)-th iteration step. For the \(j\)-th problem in Equation (14), the formulation of the shifting vector is explained as in Figure 2. For convenience of presentation, the constraint contains only two evidence variables \(\boldsymbol{Z}=(a,\,b)\). \(F\) represents the domain of \(f\leq v_{j}^{(k-1)}\).