## 2 Introduction to Optimized Method
### Multi-Objective Programming Model for Cable Force Optimization
#### 2.2.2 Optimization Objectives  
Cable-stayed bridges belong to the category of complex bridge systems. Typically, they form a triangular structure through the interaction between the stay cables, pylon, and girder to support loads. The main girder primarily resists bending moments and compression forces, while the stay cables are subjected to tensile forces. The bridge pylon mainly bears compressive forces. Figure 1 provides a typical force diagram of a cable-stayed bridge.  
Based on the deformation and mechanical characteristics of a cable-stayed bridge with a steel box girder during construction, the vertical displacement for girder and the horizontal displacement for pylon \(\delta_{i}(i=1,2,\cdots,l)\) (where \(l\) represents the total number of nodes of the girder and pylon), bending moment \(M_{i}(i=1,2,\cdots,m)\) (where \(m\) the total number of elements of girder and pylon), and cable force \(T_{\rm k}(k=1,2,\cdots,n)\) under dead load can be taken as control parameters. If the target values of each control parameter in the reasonable completed state are \(M_{0i}(i=1,2,\cdots,m)\), \(\delta_{0i}(i=1,2,\cdots,l)\), and \(T_{0k}(k=1,2,\cdots,n)\), then the differences between the current state and the reasonable completed state can be calculated using the forward-iteration method. The vector form of these differences is as follows:  
\[\left\{\begin{array}{l}\Delta\mathbf{M}=\mathbf{M}_{0}-\mathbf{M}\\ \Delta\xi=\delta_{0}-\delta\\ \Delta\mathbf{T}=\mathbf{T}_{0}-\mathbf{T}\end{array}\right., \tag{2}\]  
where \(M_{0}\), \(\delta_{0}\), and \(T_{0}\) represent the vector of the bending moment, the displacement (vertical displacement for the girder and the horizontal displacement for the pylon) vector of the girder and pylon, and the cable force under reasonable completed state, respectively. \(\mathbf{M}\), \(\delta\), and \(\mathbf{T}\) represent the vectors of the bending moment, displacements of the girder and pylon, and cable forces under the current state as calculated by the forward-iteration method. \(\Delta\mathbf{M}\), \(\Delta\delta\), and \(\Delta\mathbf{T}\) represent the difference vectors of bending moments, displacements of the girder and pylon, and cable force, respectively, between the reasonable completed state and the current state.  
The basic idea of this method is to minimize the difference of each control parameter's target value and its current state. Based on the concept of influence matrix, the optimization objectives can be formulated as follows:  
\[\left\{\begin{array}{l}\mathbf{M}(\Delta\mathbf{N})=\mathbf{C}_{\rm M}\Delta\mathbf{N}- \Delta\mathbf{M}\\ \delta(\Delta\mathbf{N})=\mathbf{C}_{\rm S}\Delta\mathbf{N}-\Delta\delta\\ \mathbf{T}(\Delta\mathbf{N})=\mathbf{C}_{\rm T}\Delta\mathbf{N}-\Delta\mathbf{T}\end{array}\right., \tag{3}\]  
where \(\mathbf{C}_{\rm M}\), \(\mathbf{C}_{\rm S}\), and \(\mathbf{C}_{\rm T}\) represent the influence matrices for bending moment, displacement of the girder and pylon, and cable force, respectively, under unit cable force.  
Figure 1: Typical force diagram of a cable-stayed bridge.