## 1 Introduction
A support vector machine was utilized for constraint verification, and the presented method was proven to be accurate and computationally efficient for prediction purposes. [PERSON] et al. [7] obtained the optimum design of prestressed concrete bridges. Design variables included the number of prestressing tendons, cable forces, cable areas, and girders' and towers' sectional dimensions. [PERSON] et al. [8] examined multi-span cable-stayed bridges while considering geometric nonlinearity effects. The variables included anchorage positions, cable forces, and cable section areas. The SQP algorithm was utilized to minimize the total cost of steel, and sensitivity analysis was conducted using the finite difference method. However, for the implementation of optimizing a practical cable-stayed bridge, handling all types of variables simultaneously is typically not the most effective strategy, because this formula can significantly lead to the increase in the dimensionality, nonconvexity, and nonlinearity of the problem. Moreover, it should be noted that optimizing the distribution of internal forces is not intrinsically contradictory to the optimization of the cost. Taking advantage of this feature, the optimization can be separated into a hierarchical layout, i.e., sequentially optimizing the mechanics-related variables and the other ones. Following this idea, [9] is one of the earliest works that introduced surrogate functions that reveal the potential connection between mechanical variables and other sizing and geometrical variables, therefore decoupling the cable forces optimization from the structural design optimization. To develop the functions, polynomial regression with the ordinary least square method was adopted. The necessary data for the regression were collected from a large parametric study conducted by repeating the finite element technique, while varying three parameters. The surrogate functions, expressed as quadratic polynomials, explicitly related cable forces to three variables concerning the span length, the total length, and the upper structure height. With these functions, the cable forces can be easily determined for different variable values to achieve the optimum post-tensioning distribution, which can minimize the deflection in both the deck and pylon. In later research [10], these functions are used to facilitate the optimum design of a cable-stayed bridge, considering variables such as the cables' section areas, prestressing forces, and cross-sectional dimensions of the girder. Although the post-tensioning functions were assessed as accurate in the previous investigation [10], there are still drawbacks when it comes to optimum design problems with more complex design conditions. Firstly, the variables included in the functions are limited in terms of the number and the types, which may lead to inaccurate predictions for the bridges if some of the other design conditions are changed. Secondly, the variables of the surrogate functions are supposed to be consistent with those of the optimum design problem. Therefore, there is a demand for reconstructing the surrogate function if some new variables that may have a visible influence on the post-tensioning distribution are introduced. Thirdly, as the number of design variables increases, the nonlinearity of the problem significantly grows. The polynomial regression will be insufficient to give accurate predictions, due to its poor anti-interference and local fitting ability, compared with other regression methods. To this end, machine learning techniques have recently garnered significant attention. Researchers have explored various tools, including Random Forest [11], Support Vector Machines [12], and Physics-informed or Data-driven Neural Networks [13], to predict structural performance or responses. In a broad sense, as these tools are strategically designed as a cheaper-to-evaluate substitute for the original sophisticated computational model, they are also typically called surrogate models or meta-models [14].