## 2 Introduction to Optimized Method
### Optimization Process for the Reasonable Construction State of Cable-Stayed Bridge with Steel Grider  
Based on the above analysis, the optimization process for the construction state of a cable-stayed bridge with a steel box girder using the multi-objective programming method can be summarized as follows:  
1. Establish the finite element model of the cable-stayed bridge with the steel box girder based on the forward-iteration method. Calculate the influence matrices for displacement \(\mathbf{C}_{\mathcal{B}}\) and bending moment \(\mathbf{C}_{\mathrm{M}}\) of the girder and pylon, and cable force \(\mathbf{C}_{\mathrm{T}}\) under unit cable force according to the actual construction process.
2. Use the cable force under the reasonable completed state as initial iterative cable tension \(\mathbf{N}^{(1)}=\mathbf{T}_{0}\) to conduct forward-iteration calculation and make \(k=1\); then, calculate vectors of bending moment \(\mathbf{M}^{(k)}\), displacement \(\mathbf{\delta}^{(k)}\) of girder and pylon, and cable force \(\mathbf{T}^{(k)}\) under current completed state. The difference vector for each control parameters between the reasonable completed state and the current completed state \(\mathbf{d}^{(k)}=\left(\Delta\mathbf{M}^{(k)},\Delta\mathbf{\delta}^{(k)},\Delta\mathbf{T}^{( k)}\right)^{\mathrm{T}}\) can be determined.
3. Based on Equation (4), a multi-objective programming model can be established by taking \(\mathbf{f}\!\!\left(\Delta\mathbf{N}^{(k)}\right)=\mathbf{C}\Delta\mathbf{N}^{(k)}-\mathbf{d}^{( k)}\) as the optimization objective.
4. Convert the multi-objective programming model into an unconstrained single-objective programming model using the weighted sum of squares method.
5. Provide a permissible error \(\varepsilon>0\). Using MATLAB and the damped Newton method, the adjustment amount for cable tension \(\Delta\mathbf{N}^{(k)}\) can be determined.
6. Use \(\mathbf{N}^{(k+1)}=\mathbf{N}^{(k)}+\Delta\mathbf{N}^{(k)}\) as the updated cable tension and perform the forward-iteration calculation. A new set of control parameters under the current state, along with the updated differences between the current state and the reasonable completed state \(\mathbf{d}^{(k+1)}=\left(\Delta\mathbf{M}^{(k+1)},\Delta\mathbf{\delta}^{(k+1)},\Delta\mathbf{ T}^{(k+1)}\right)^{\mathrm{T}}\) can then be obtained.
7. Provide a permissible error \(\varepsilon_{1}>0\) and \(\varepsilon_{2}>0\). Verify whether both formulas \(\sum\limits_{i=1}^{\mathrm{n}}\left|\Delta N_{i}^{(k)}/nT_{i}\right|\leq \varepsilon_{1}\) and \(\max\left|\Delta N_{i}^{(k)}/T_{i}\right|\leq\varepsilon_{2}\) are valid. If they are, terminate the iteration. The optimal cable tension during the construction process \(\mathbf{N}^{*}=N^{(k+1)}\) can be determined. At this stage, the reasonableness of the construction state should be evaluated based on the forward-iteration results. If the calculation results are deemed reasonable, the corresponding reasonable constructing state and reasonable  
Figure 2: Iteration process of damped Newton method.  
completed state are obtained. If the results are not reasonable, adjust the objective functions or weight coefficients and return to step (3) to rebuild the optimization model and continue the iteration process. This procedure continues until the error meets the specified precision requirements. The implementation process is illustrated in Figure 3.