## 3 Definition of Optimization Problem
### Objective Functions  
Figure 5 shows the internal force change of the crossing cable under the action of horizontal force \(\delta H\). Under the horizontal force \(\delta H\), the crossing cable force of the mid tower increases, the crossing cable force of the side tower at the same anchor point decreases, and the allocation of the deck' dead load between the crossing cables changes, thus limiting the deformation of the mid tower.  
[PERSON] et al. [26] derived the analytical formula for the constraint rigidity of the cross cables on the towers based on the interaction mechanism of crossing cables:  
\[K=\frac{E_{d_{r}}}{l_{l}^{3}/A_{l}+l_{r}^{3}/A_{r}} \tag{5}\]  
where, \(E\), \(A\) and \(l\) are the elastic modulus, are and length of the crossing cable, and \(a_{r}\) is the length from the anchorage position of the crossing cable on the deck to the mid tower. To clarify Equation (5), the terminology used is shown in Figure 5.  
As can be seen from Equation (5), the constraint stiffness of the crossing cable is related to its area, anchorage position and elastic modulus. When the stress level of the crossing cable under dead load is \(\sigma\)[27], the area \(A\) of the crossing cable can be expressed as:  
\[A=\frac{F}{\sigma} \tag{6}\]  
where \(F\) is the pre-tension force of the crossing cable.  
Substituting Equation (6) into Equation (5), constraint stiffness of crossing cables is expressed as:  
\[K=\frac{1}{\sigma}\cdot\frac{E_{d_{r}}}{l_{l}^{3}/F_{l}+l_{r}^{3}/F_{r}} \tag{7}\]  
According to Equation (7), when the stress level under dead load and anchorage position of the crossing cable are constant, the constraint stiffness of the crossing cable is only related to the pre-tension forces. The allocation ratio of the deck' dead load among the crossing cables determines the constraint stiffness of the crossing cables on the mid tower. In this study, the stress level under dead load is 590 MPa, and the crossing cable area is calculated by Equation (6). The stress level does not affect the allocation ratio of the deck' dead load between the crossing cables.  
Reducing the deflection of the deck caused by live load is the key problem of multi-tower cable-stayed bridge. Therefore, the optimization objective is defined as minimizing the displacement of the tower and beam under dead loads and minimizing the displacement of the dead under live loads. This definition ensures that the line shape of the tower and deck is smooth under dead load, and the structural performance is optimal.  
Figure 5: Interaction mechanism of crossing cables.  
The first objective function is the square root of the sum of squared displacements of nodes between the deck and tower under dead load. \(F_{obj1}\) is formulated as follows:  
\[F_{obj1}=\sqrt{\left(\delta_{1}^{2}+\delta_{2}^{2}+\ldots\right)+\left(\Delta_{ 1}^{2}+\Delta_{2}^{2}+\ldots\right)} \tag{8}\]  
where \(\delta_{i}\) is the vertical displacement of the deck nodes within the range from the end of the side span to the mid tower, and \(\Delta_{i}\) is the horizontal displacement of the side tower nodes.  
The second objective function is the cube root of the ratio of the sum of the maximum upward and downward displacement of the deck to the span length under live load. \(F_{obj2}\) is formulated as follows:  
\[F_{obj2}=\sqrt[3]{\left(\delta_{max}+\delta_{min}\right)/L} \tag{9}\]  
where \(\delta_{max}\) and \(\delta_{min}\) are the maximum upward and downward displacement of the deck caused by live load.  
The linear weighting method is used to transform the two optimization objectives into a single objective problem [28]. By introducing the weighting coefficient \(\alpha_{i}\) to combine the two objective functions, the objective function \(F_{obj}\) is obtained:  
\[F_{obj}=\alpha_{1}F_{obj1}+\alpha_{2}F_{obj2} \tag{10}\]  
The weighted coefficient \(\alpha_{i}\) reflects the relative importance of each objective function in the optimization process. By adjusting the weighting coefficient, the emphasis of the optimization results can be controlled.