## 1 Introduction  
A cable-stayed bridge is a composite structural system, primarily composed of cables, pylons, and main girders [1], which has been established it as one of the most competitive bridge types in modern bridge engineering due to its slender and aesthetically pleasing form, combined with its high strength, low self-weight, and significant spanning capacity [2, 3]. Cable-stayed bridges with steel box girders, characterized by high structural integrity and efficient construction, are favored by engineers. Recent years have seen increased traffic volumes, leading to the wider application, larger spans, more slender main beams, and greater variety of the structural systems in cable-stayed bridges with steel box girders. Simultaneously, determining a reasonable finished state has become a critical aspect of the design and analytical calculations for cable-stayed bridges [4; 5]. Accurate and efficient determination of a reasonable finished state is therefore crucial for the application and development of these bridges. The design process for cable-stayed bridges with steel box girders aims for a reasonable finished state, characterized by a uniform distribution and a smooth profile of the internal forces in the main girder and pylons under the permanent loads of the erected structure and a uniformly increasing cable force as a function of the cable length. Structural layout and cable forces are the primary factors influencing the reasonable finished state of cable-stayed bridges. Cable forces also influence the structural layout, and adjusting the cable forces is relatively easier. Therefore, cable forces are generally chosen as the optimization objective. During the design of the Swedish Stromsound cable-stayed bridge with steel box girders [6], [PERSON] considered cable force calculations for the first time, and cables transitioned from passive load-bearing elements to components capable of actively adjusting the internal force distribution. Optimizing the cable tension allowed for the attainment of reasonable internal forces and profiles in the finished state. This demonstrates that given the structural system and the dead load distribution of a cable-stayed bridge, the problem of optimizing the finished state of a cable-stayed bridge can be transformed into an optimization problem focused on the cable forces [7; 8]. Iterative development has led to diverse cable force optimization methods for cable-stayed bridges. The common approaches fall into four categories. The first category comprises optimization methods for specified structural states, primarily including the rigid support continuous beam method [9], the zero-displacement method, and the internal force (stress) equilibrium method [10; 11]. While each of these methods has its advantages, they fail to fully account for the stresses and deformations of all of the components in the cable-stayed bridge [12]. The second category employs the minimum bending energy (moment) method [13]. This approach does not target a predetermined structural state but rather optimizes the cable forces by minimizing the structure's bending strain energy. Notwithstanding capturing the essence of cable force optimization, its capacity is confined to optimizing the cable forces in the dead load state of the completed bridge, excluding considerations of prestress and live loads. The third category utilizes the influence matrix method [14; 15]. This approach establishes a functional relationship between the cable forces and optimization objectives through influence matrices. However, relying solely on the influence matrix method does not clearly define the optimization target and often requires a large number of iterations. The fourth category employs mathematical optimization methods [16; 17], primarily linear and nonlinear programming algorithms. Linear programming algorithms still fail to account for the states of all of the bridge components.