## 6 Conclusions
The predictor addresses the time-consuming problem of determining the optimum cable prestressing forces in each of the iteration rounds in the optimization problem. This predictor is expected to be the highlight of this paper. * to deal with the nonconvex optimization problem, an optimization program consisting of the initialization module, iteration module, and structural analysis module is made based on PSO algorithm. PSO is well-known for its global searching ability. The global searching ability is enhanced by the following measures: a moderate population number of 30 which is five times to the number of the design variables; well-set defining parameters of the algorithm; and the dual strategy (refer to Appendix B) adopted when initializing particles. Finally, the optimum design of a (\(358+818+358\))-meter-long cable-stayed bridge was chosen as the implementation of the proposed methods. First and foremost, the theoretical material cost was set as the objective function. Six design parameters closely connected with cost of the towers and the cables were identified as design variables. Constraints were set consistently with the Chinese Design Code. Then, 972 samples were generated by uniformly discretizing the design variables. After training and comparison, Gaussian Process Regression was demonstrated as the best surrogate model for prediction. The cable forces predictor was successfully constructed when GPR passed the testing. Afterward, the optimization program was launched to obtain the optimum design of the variables. The cable forces predictor was utilized to determine the optimum cable tensions and counterweight in the program. The predictor eliminated the need to solve the cable forces optimization problem in each of the iteration rounds, resulting in improved efficiency. The results show that the theoretical material cost of the optimum design is 32% lower than the original design. The feasibility and reliability of the structural optimization process were verified by several checks. Despite our proposed method demonstrated accuracy and efficiency in our example, there are still some possible improvements when confronting larger scale problems. For instance, the surrogate model is naturally a regression trained with given samples, which indicates that if some new design variables are introduced, or if a wider searching range is to be explored, the model needs to be updated for the sake of accuracy. Therefore, it is beneficial to construct a larger version of the cable forces predictor covering a broader range of potential design variables for cable forces optimization problems. In addition, the sampling strategy and different surrogate models can also be more comprehensively compared. These directions can be potentially investigated in future work. Conceptualization, Y.M. and C.S. ; methodology, Y.M. and C.S. ; software, Z.W. ; validation, Z.W. ; formal analysis, Y.M. ; investigation, C.S. ; resources, R.X. ; data curation, Z.W. ; writing--original draft preparation, Y.M. ; writing--review and editing, Y.M. and C.S. ; visualization, Z.J. ; supervision, R.X. ; project administration, B.S. ; funding acquisition, B.S. All authors have read and agreed to the published version of the manuscript. **Funding:** This research was funded by National Key R&D Program of China, grant number 2021 YFB1600300, National Natural Science Foundation of China, grant number 52378185, and the Science and Technology Committee of Shanghai, China, grant number 21 DZ1202900. The opinions and statements do not necessarily represent those of the sponsors. **Institutional Review Board Statement:** Not applicable. **Informed Consent Statement:** Not applicable. **Data Availability Statement:** The data presented in this study are available on request from the corresponding author. The data are not publicly available due to privacy. **Conflicts of Interest:** Author [PERSON] was employed by the company CCCC Highway Consultants Co., Ltd.