## 1 Introduction  
Cable-stayed bridges have been widely constructed to span roads and rivers. As a highly redundant structure, cable-stayed bridges have advantages in terms of the stiffness, wind-load resistance, maintenance, and span-crossing ability. As a result, cable-stayed bridges have exceeded a span of 1 km in just 60 years since the first cable-stayed bridge was built. The Chang-Tai Yangtze River Bridge in China, with a total length of 5.3 km and a main span of 1176 m, is the world's longest cable-stayed bridge for both highway and railway. The construction of such a huge engineering structure is extremely challenging and costly. Therefore, it is becoming increasingly critical to carry out detailed structural optimization and comparison at the design stage. For a practical engineering project, the implementation of bridge design optimization can face a couple of challenges. First, the involved design parameters for optimization of a cable-stayed bridge are typically high-dimensional. As a highly redundant structure, there are numerous design parameters to be optimized in the bridge design process. Variables involved can be classified into mechanical, sizing, geometrical, and topological [1]. The optimization problem becomes highly dimensional due to the increasing number of variables, which hinders efficiency and consumes more time. The interdependency among design variables exacerbates nonconvexity and nonlinearity, making the problem difficult and challenging. In addition, various constraints need to be considered in the optimization design of cable-stayed bridges. Due to the complexity of real-world environments and bridge-operating conditions, engineers should consider the varieties of site, structure, material, member, section type, load cases, and the complex verification items of specification. These considerations, whether linear or nonlinear, make the feasible domain very limited within the design space. Moreover, the mechanical behavior of cable-stayed bridges must be properly simulated. Cable-stayed bridges transmit loads with cables. They are relatively flexible structures compared to girder and arch bridges. Therefore, the geometric nonlinearity caused by sag effect of cables, large displacement effect, and \(p\)-\(\Delta\) effect, should be considered. Though current commercial software can handle these non-linearity effects by finite element analysis, the simulation can be time-consuming and undermine the feasibility of optimization. To address these challenges, a lot of research has been conducted to advance the computational methods or strategies for the design optimization of cable-stayed bridges. Early studies conducted by [PERSON] [2] introduced an optimality-criteria-based method to determine the prestressing forces of cables in steel bridges. Similar early studies involved plenty of assumptions and addressed the problem using simplified mathematical formulas. [PERSON] et al. [3] minimized the total strain energy expressed as a quadratic function of the post-tensioning cable forces with an influence matrix. [PERSON] et al. [4] optimized cable areas for a long-span steel bridge with the finite differences sensitivity analysis method and solved the problem through a gradient-based sequential quadratic programming algorithm. A three-stage algorithm was presented by [PERSON] et al. [5] to optimize the cable prestressing tensions with a nonlinear inelastic analysis. Besides the above work on optimizing single type of variables, research has also been carried out to pursue a more comprehensive \"optimum design\" of the bridge. The structural design problem is formulated with various variables including not only mechanical but also sizing, geometrical, and topological variables. [PERSON] et al. [6] proposed an optimization method for cable-stayed bridges that utilized a genetic algorithm to minimize costs while considering geometrical parameters and cross-sectional dimensions as design variables.