## 2 Problem Formula  
In a general cable-stayed bridge design, engineers need to carry out comprehensive design and optimization from the structure configuration to the detail members. In terms of the configuration, crucial parameters including the length of the main and the side span, the width of the deck, the height of the towers, and the anchorage position of the cables should be well determined. In terms of the members, cross-sectional dimensions of the main members including the towers, the cables, and the deck are supposed to be well designed. The typical optimum design formula with all design variables optimized altogether can be stated as  
\[ \begin{split}\min_{\mathbf{X}\mathbf{Q}}\Phi&= \sum_{j=1}^{m}C_{j}V_{j}\\ s.t.x_{il}&\leqslant x_{i}\leqslant x_{iu}\text{ for all }i \in N_{X}\\ q_{kl}&\leqslant q_{k}\leqslant q_{ku}\text{ for all }k \in N_{Q}\\ \sigma_{jl}&\leqslant\sigma_{j}^{(p)}\leqslant \sigma_{ju}\text{ for all }j\in M,\text{ all }p\in L\\ f_{j}^{(p)}&\leqslant f_{ju}\text{ for all }j\in M, \text{ all }p\in L\\ K_{j}^{(p)}&\leqslant K_{ju}\text{ for all }j\in M, \text{ all }p\in L\end{split}  \tag{1}\]  
where \(\Phi\) represents the total cost of members, \(\mathbf{X}=\{x_{1},x_{2},x_{3},\ldots,x_{n_{X}}\}\) stands for sizing (e.g., cross-sectional dimensions) and geometrical (e.g., side-span ratio, height-span ratio) design variables, and \(\mathbf{Q}=\left\{q_{1},q_{2},q_{3},\ldots,q_{n_{Q}}\right\}\) stands for mechanical (e.g., cable prestressing forces, tendon prestressing forces in reinforced concrete beams) design variables. \(C_{k}\) and \(V_{k}\) represent the cost coefficient and the volume of the \(j\)-th member, respectively. \(M=\{1,2,3,\ldots,m\}\) is the member number set. The next two constraints represent the lower and upper boundaries of the \(i\)-th variable in \(\mathbf{X}\) and the \(k\)-th variable in \(\mathbf{Q}\), where \(N_{X}=\{1,2,3,\ldots,n_{X}\}\) and \(N_{Q}=\left\{1,2,3,\ldots,n_{Q}\right\}\). The following three constraints represent, respectively, the strength, stiffness, and stability constraints of the structure. The first constraint of these represents the upper bound of the \(j\)-th member's stress response, where \(p\) is the load case identifier and \(L=\{1,2,3,\ldots,l\}\) is the load case number set. The second represents the boundary of the deflection response, and the third represents the boundary of the stability coefficient. It is worth noting that wind or seismic resistance is also critical, especially in the design of long-span bridges; however, due to the limit of the paper length, such dynamic performance is not discussed in this paper. Therefore, the girder sizes are excluded from \(\mathbf{X}\) and not optimized in the later example. Meanwhile, considering that cable prestressing forces are more dominant than tendon prestressing forces in preliminary design, \(\mathbf{Q}\) consists only of cable forces in the later example. In this optimization, \(\mathbf{X}\) and \(\mathbf{Q}\) are variables optimized together, where \(n_{q}\) is generally much larger than \(n_{X}\) due to the high density of cables. However, an optimization process dealing with \(\mathbf{X}\) and \(\mathbf{Q}\) simultaneously is highly complex and computationally costly, due to their nonlinear and conflicting (i.e., coupling) feature [1]. Thus, in this work, a reasonable two-layer framework is adopted to decouple them into two sub-problems in one optimization round. Sub-problem one (Sp1) is an internal force distribution optimization problem where \(\mathbf{Q}\) is optimized with fixed \(\hat{\mathbf{X}}\), as most of the \"cable forces optimization\" problems [18; 19]. The most common and practical objective function of Sp1 is the weighted sum of members' strain energy. The formula of Sp1 can be stated as  
\[ \begin{array}{c}\underset{\mathbf{Q}}{\min}U=\quad\underset{\mathbf{Q}}{ \min}\int_{S}\frac{M^{2}(S,\mathbf{Q})}{2 EI}dS\\ q_{kl}\leqslant q_{k}\quad\leqslant q_{ku}\text{ for all }k\in N_{Q}\\ f^{(deadload)}\quad\leqslant f_{u}\quad\text{for all }p\in L\end{array}