## 1 Introduction  
A cable-stayed bridge represents a composite structural system comprising tensioned cables, compressed towers, and beams subjected to both bending and compression [1]. Owing to their favorable structural resilience, aesthetical design, and well-established construction techniques, they are widely used [2]. The development of cable-stayed bridges has been traced from 1784 to contemporary times, and the first modern cable-stayed bridge, the Stromsend Bridge, was completed in 1955 in Sweden [3]. Since the construction of the first modern bridge of this type, there has been a noticeable rise in both the number of constructed cable-stayed bridges and the lengths of their spans [4]. Over the years, cable-stayed bridges have gained significance as their characteristics have been comprehensively explored. They have recently demonstrated exceptional cost efficiency for spans ranging from the short, medium, to long lengths [5]. Consequently, research on all aspects of the design, structural analysis, construction, inspection, maintenance, damage detection, and rehabilitation of cable-stayed bridges has emerged as a contemporary and relevant focus within the field of bridge engineering [6]. The studies [7; 8; 9; 10], among others, have significantly advanced the development of computational models for cable-stayed bridges, leading to a better understanding and design of these structures in terms of their static and dynamic performance. Cables function as the primary vertical load-bearing and force-transmitting components of the bridge, and even minor fluctuations in cable tension can significantly impact the overall behavior of other bridge elements, such as the deck and pylon. The structural integrity of cable-supported bridges is reliant upon steel cables, which are employed to support both the bridge deck and its associated load [11]. Therefore, determining and optimizing the initial forces in the cables is of crucial importance. Given that the determination of initial cable forces within the structural analysis is a crucial initial stage in assessing the performance of the bridge for external loads (which are not directly the subject of this study) and precedes the application of designed forces onto the cable, only a brief overview of both classical and newer methods for determining and optimizing these forces will be provided. There are several common approaches for determining initial cable forces. In 1993, [PERSON] et al. [12] pioneered the zero-displacement method for determining post-cable tension in cable-stayed bridges subjected to a dead load. However, this approach fails to consider the significant horizontal displacement of girders and towers when the structure is subjected to asymmetrical loading. Based on the force equilibrium approach, [PERSON] et al. [13] developed a novel methodology for determining cable forces in cable-stayed bridges. Despite its greater efficiency compared to previous methods, this approach also overlooks the final shape of the towers and girders. The other traditional methods that need to be mentioned are the minimum bending energy method, the rigid supported continuous beam method, and the influence matrix method [14]. In recent decades, there has been an expansion of cable force optimization methods. These methods employ various mathematical optimization algorithms to efficiently distribute forces within cable-supported structures such as bridges. The primary objectives are to minimize material usage, reduce construction costs, and simultaneously enhance structural stability and safety. A detailed overview on the optimization of cable-stayed bridges is provided in a literature survey conducted by [PERSON], [PERSON] et al. [15].