## 1 Introduction  
Cables are an important component of cable-stayed bridges, since they bear most of the gravity load of the bridge [1, 2]. Due to environmental corrosion, wind- and rain-induced vibration, material fatigue, and other adverse factors, cable force may be damaged with the increase in service life [3, 4]. The local damage in the cable may lead to rapid damage of the whole bridge structure. It is necessary to find local damage in cables before failure occurs. Generally, the wires of the cables are protected by PE pipe. Damage identification through observation requires dissection of the PE pipe, which is inapplicable for long-term health monitoring [5]. The image-based approach is only suitable for surface damage to the cable [6]. Other damage assessment methods [7, 8], such as impulse radar and thermography, usually require expensive testing equipment. In recent years, vibration-based damage identification methods [9, 10, 11, 12] have attracted extensive attention in the field of bridge engineering. Changes in structural vibration frequency and mode shape often indicate the occurrence of structural damage. For beam-type structures, [PERSON] et al. [13] used vibration frequency and mode shape change to construct a new damage indicator. Using a statistical technique, [PERSON] and [PERSON] [14] developed a damage assessment method using time series data. In reference [15],the normalized uniform load surface (NULS) curvature was defined and used to detect damage in beam structures. It was found that the NULS curvature only changed in the damaged area. [PERSON] et al. [16] combined the eigenvector curvature with the eigenvalue of structural vibration changes to identify damages in a conical shell. For cable-stayed bridges, [PERSON] et al. [17] used a modal strain energy method to find structural damages. In reference [18], the wavelet packet transform and the neural network were combined to assess possible damages in structures. In reference [19], an optimal algorithm based on elementary energy was proposed to determine the damage in a steel structure. Based on a moving load, a wavelet-based technique [20] was developed to detect beam crack. In reference [21], a new algorithm was proposed to identify the damage to bridge structures by using the Hilbert-Huang transform. In reference [22], the moving load response surface method was developed to evaluate the safety conditions of a bridge structure. [PERSON] et al. [23] presented a wavelet-transform-based method for structural damage identification by comparing the discrete wavelet transforms of both the undamaged and damaged structures. [PERSON] et al. [24] analyzed the dynamical behavior of damaged cables and investigated damage detection based on vibration frequency measurement. Although the research of vibration-based methods has made great progress, there are still many problems in its application in engineering practice. In reference [25], the main disadvantages of vibration-based methods were pointed out as follows: (1) Structural damping had a direct impact on structural vibration data; however, many vibration-based methods do not consider the influence of structural damping. (2) For large civil engineering structures, it was difficult to excite the vibration of the structure by artificial methods. (3) The current modal testing technology presents difficulties in measuring and obtaining the high-order modal data of structural vibration. In practice, static damage identification methods are only related to structural stiffness, and static data can be easily and accurately measured in practice. Therefore, structural damage identification methods based on static data have been more widely used in the field of civil engineering. Using static displacement, [PERSON] and [PERSON] [26] proposed an iterative optimization method to identify the stiffness parameters of each element in a structure. [PERSON] et al. [27, 28] summarized various damage identification optimization algorithms based on static data and pointed out the existing problems to be overcome.