## 1 Introduction  
Since the mid-20 th century, there have been significant advances in cable-stayed bridge technology [1], especially in multi-tower design, to enhance the bridge's ability to span wide waters [2]. The Rion-Antirion Bridge, the Millau Viaduct and the Jiashao Bridge are examples of this technological development. With the increase of the number of towers, how to improve the longitudinal stiffness of mid towers has become a key problem. For the first time, the Queensferry Crossing adopted crossing cables to limit the deflection of the tower, thereby enhancing the structural rigidity [3]. The zone where the deck is supported by the crossing cables is called the crossing zone, and the dead load of deck is transferred to the tower on both sides through the crossing cable. The pre-tension forces of crossing cable not only determines the proportion of the tower bearing the deck' dead load, but also affects the selection of the area of the cable. The diameter of the cable increases with the increase of the pre-tension force, which is directly related to the performance of the crossing cables. Therefore, the pre-tension forces of crossing cable is the key in the design of multi-tower cable-stayed bridges. The following is an overview of several early proposed methods for cable force optimization. [PERSON] et al. [4] proposed the zero displacement method, which optimizes structural performance by adjusting cable forces to make the displacement of nodes on the deck approach zero. [PERSON] et al. [5] introduced the Kriging model into the zero displacement method, reducing the number of iterations. [PERSON] et al. [6] proposed an optimization method based on force balance, which takes the bending moment of the deck as the optimization objective, and continuously adjusts the force applied to the anchor point to find the cable force distribution that achieves the desired bending moment. [PERSON] et al. [7] proposed the unit load method, which considers the construction process and time-dependent effects in the cable force optimization process. [PERSON] et al. [8] proposed a practical method based on the cable force influence matrix, aimed at minimizing the bending energy in the tower and deck to find the optimal cable forces. With the extensive application of computational methods such as artificial intelligence in engineering [9], the cable force optimization problem has ushered in a new solution strategy. [PERSON] et al. [10] emphasized the importance of post-tensioning cable forces for improving the structural performance, especially in solving the unbalanced load caused by dead load. Then, the post-tensioning cable forces are taken as the design variables, and the objective function is established based on the minimum strain energy theory to calculate the optimal post-tensioning cable forces. [PERSON] [11] used finite difference method and sequential quadratic programming algorithm to optimize crossing cable system of Queensferry Crossing, effectively reducing the steel volume in cables. [PERSON] et al. [12] considered the counterweights acting on the side spans during the cable force optimization process, and reduced the bending energy in the main girder and towers by adjusting the counterweights and cable forces. Subsequently, [PERSON] et al. [13] took the length of the counterweight and the bending energy of the tower and deck as the objective function, and solved the dual objective optimization problem by improving the Pareto weighting coefficient method. The results show that the force of the structure under dead load permanent state can be improved by considering the action length of the counterweight. [PERSON] et al. [14] first applied B-spline curves to represent the distribution of post-tensioning cable forces along the bridge length, greatly reducing the number of design variables. And B-spline curves were used to represent the distribution of cable area, optimizing the steel consumption of cables [15]. [PERSON] et al.