## 8 Conclusions  
Using the Xiangshan Port cable-stayed bridge with steel box girders as a case study, this article investigates the optimization methods for the finished state. Through introducing constraint conditions, determination of a reasonable finished state is formulated as a mixed-constraint quadratic programming problem, which is then solved using the multiplier path following algorithm. Furthermore, the influence of the constraint ranges on the optimization results is analyzed, considering the vertical displacement of the main girder, the horizontal displacement of the piers, the cable forces, and cable force uniformity. The principal conclusions are summarized below:  
(1) Convergence of the objective function is achieved within 30 iterations of the mixed-constraint quadratic programming model. The optimized profile of the main girder is smooth, with a uniform stress distribution along the top and bottom flanges; the top of the pylon exhibits a predetermined lateral offset towards the outer span, with a fully compressed cross-section; the cable forces increase uniformly with cable length; and positive vertical support reactions are observed in both transition and auxiliary piers, indicating sufficient pressure reserves. This suggests the reasonableness of the constraint range and the feasibility and efficiency of the optimization method for the finished state.  
(2) The multiplier path following optimization algorithm is computationally efficient, exhibiting good convergence and insensitivity to the problem size. Being easy to program, it avoids the arbitrariness of manual cable adjustments, enabling straightforward determination of the reasonable finished state for a cable-stayed bridge with steel box girders. The optimized results provide a theoretical benchmark for the design of cable-stayed bridges with steel box girders.  
(3) Regarding the Xiangshan Port cable-stayed bridge with steel box girders, the vertical displacement of the main girder, the positive and negative bending moments, and the normal stresses at the top and bottom edges, as well as the positive and negative bending moments in the towers, are significantly influenced by the constraint ranges. The horizontal displacement of the pylon roof is significantly affected by the constraint ranges of both the main girder's vertical displacement and the pylon's horizontal displacement, while the remaining constraint ranges have a limited impact.  
Conceptualization: J.S., Y.T. and J.D. ([PERSON]). Methodology: J.S., Y.T. and F.Q. Software: J.S. and J.D. ([PERSON]). Validation: J.S. and Q.X. Investigation: Y.T. and Q.X. Data curation: Q.X. and J.D. ([PERSON]). Writing--original draft preparation: J.S., Y.T. and F.Q. Writing--review and editing: J.S., Y.T. and F.Q. Funding acquisition: J.D. ([PERSON]). All authors have read and agreed to the published version of the manuscript.  
The authors express their sincere gratitude to the National Natural Science Foundation of China (52192663) and for the financial support provided by the National Key R&D Program of China (2021 YFF0501004).  
Not applicable.  
**Data Availability Statement:** The original contributions presented in this study are included in the article; further inquiries can be directed to the corresponding authors.  
**Acknowledgments:** The authors thank the reviewers for their great help with the article during the progress of its review.  
**Conflicts of Interest:** Author [PERSON] and [PERSON] were employed by the China Harbour Engineering Company Limited. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.