## 3 Mathematical Model of Cable Force Optimization
### Objective Function  
At present, the accepted principle for determining the reasonable completion state of cable-stayed bridge structure is [20]:  
1. To maintain the vertical state of the bridge when the bridge is formed, and to reduce the horizontal displacement and meet the limit requirements without excessive local bending moment at the base of the bridge tower. 2. To maintain the horizontal state of the girder when forming a bridge, to reduce its vertical displacement and to meet the limit value. 3. To ensure the cable force distribution is uniform, which means that the cable force is positively correlated with the length of the cable itself. 4. To induce no negative force at the support. In the above principles, (1) and (2) are essentially constraints on the distribution and alignment of internal forces in the bridge state of members. The reasonable completion of the bridge state requires smooth alignment of pylons and main beams, uniform distribution of internal forces, and no excessive local stress. For the optimization objective of the main beam, the most common objective function can be the minimum bending energy of the main beam [21; 22; 23]. However, this is more applicable to equal-section beams. Since the equal-section beam has a fixed value of the flexural stiffness EI, the continuous main beam can be discretized and then solved using methods such as the influence matrix. For variable-section beams, this objective function is not applicable. Therefore, other objective functions must be sought. The main consideration is that the displacements of the beam and the main tower can reflect the bending of the main beam and the main tower to some extent, and considering that the displacement has positive and negative values, the vertical displacement of the control node of the main beam and the horizontal displacement of the control node of the bridge tower can be taken as the objective function. [PERSON] et al. [24] used the sum of the arithmetic square root of the vertical displacement of the main beam and the vertical displacement of the bridge tower as the objective function. However, the different ranges of variation of main girder and tower displacements are not reflected in it. In order to distinguish the influence weights of the bridge tower and main beam on the objective function, the main objective function is finally determined as follows:  
\[D=D_{1}+D_{2}=\lambda\sqrt{(x_{1}^{2}+x_{2}^{2}+\ldots+x_{p}^{2})}+\sqrt{(y_{1 }^{2}+y_{2}^{2}+\ldots+y_{q}^{2})} \tag{7}\]  
where \(x_{i}(i=1,2,\ldots,p)\) is the horizontal displacement of the control node of the bridge tower, n is the number of horizontal control nodes, \(y_{i}(i=1,2,\ldots,q)\) is the vertical displacement of the control node of the main beam, m is the number of vertical control nodes, and \(\lambda\) reflects that the weight of the horizontal displacement of the main tower is different from the vertical displacement of the main beam. Principle (3) is that for the cable force, the cable force distribution generally increases from the main tower to both ends (short cables to long cables). Small local jump variations in individual cable force values are allowed. Take the first pair of cable forces on both sides of the main tower and the tail cable forces as examples. The positive correlation between the cable force and its length can be explained by the fact that the longer the cable, the higher the value of the cable force, and vice versa, the lower the value. The closer the set of cable forces distributed per unit length of each cable, the more uniform the cable force distribution of the cable system. This relation is only mathematically valid because this set degree has no physical meaning and is different from the set degree of axial force in material mechanics.