## 1 Introduction
Nonlinear programming algorithms, however, surmount the limitations of the minimum bending energy method, enabling targeted selection of the objective functions, constraints, and optimization algorithms tailored to the specific characteristics of different cable-stayed bridge systems. Nevertheless, the computational process of nonlinear programming algorithms is complex, and controlling the convergence of the results is challenging [18; 19]. Accordingly, improving conventional nonlinear programming algorithms is a key path to optimizing the cable forces in cable-stayed bridges. Advances in technology have significantly aided the optimization of the cable forces in cable-stayed bridges using finite element software, but the underlying logic for solving the problem still relies on nonlinear programming algorithms. [PERSON] [20] proposed the sequential quadratic programming algorithm in 1963, with significant advancements made by [PERSON] and [PERSON] [21; 22; 23] in the 1970s. [PERSON] et al. [24] established a mathematical model based on quadratic programming, using the sum of the bending energies of the main beam and pylons as the objective function and incorporating constraints on the longitudinal compression of the main beam, the bending moments in the main beam and pylons, and the support reactions to optimize the cable forces in cable-stayed bridges. [PERSON] et al. [25; 26] used finite element programs, combined with the allocation principles of dead and live loads, to iteratively define the stresses and cross-sectional properties of post-tensioned stay cables, suspension cables, and main cables, ultimately obtaining the final layout. This procedure also considered the minimum required cross-sectional area of each stay cable under a dead load, verifying the maximum allowable stress and deflection of the cable-stayed bridge under normal and limit states. [PERSON] et al. [18; 27], utilizing the real-coded genetic algorithm (RCGA), employed B-spline interpolation curves to accurately represent the cable force distributions. They effectively addressed the substantial number of variables resulting from the large quantities of cables by calculating the cable forces in steel-concrete composite bridges through minimization of the square root of the sum of the squares (SRSS) of the deflections at the bridge deck and pylon nodes while also minimizing the weight of the steel cables to determine their optimal cross-sectional area. [PERSON] et al. [28] combined a nonlinear, inelastic analysis with genetic algorithms to simultaneously optimize the initial prestress and total weight of the cables using a unit load matrix, significantly reducing the computational costs. [PERSON] et al. [29] combined simulated annealing with cubic B-spline interpolation to select the minimum cable force from the generated objective function values as the optimal solution utilizing the OpenSees platform. The static responses of the deck, pylons, and stay cables in the cable-stayed bridge were then calculated. [PERSON] et al. [30] combined the mutation operations from particle swarm optimization (PSO) and simulated annealing (SA) with the traditional genetic algorithm (GA). By minimizing the deviation between the displacements at completion and the established construction objectives, they enhanced the algorithm's capacity to escape local minima when calculating the cable forces. [PERSON] et al. [31] employed Midas Civil and MATLAB as their structural analysis and cable force optimization tools, respectively. A multi-objective particle swarm optimization algorithm was used to construct a mathematical model for cable force optimization. Fuzzy set theory was then incorporated to determine the reasonable dead load state of the cable-stayed bridge. Synthesizing the above, the aforementioned methods have propelled the development of cable force optimization in cable-stayed bridges, but methods optimizing specified structural states fail to address the combined stresses and deformations of all of the bridge components.