## 1 Introduction  
An auxiliary pier is an important measure to enhance the overall stiffness of cable-stayed bridges [1]. The auxiliary pier can not only restrain the displacement and the bending moment of the side-span girder and reduce the stress amplitude of the stay cable [2] but can also significantly reduce the horizontal displacement of the pylon and the vertical displacement of the mid-span girder under the live load by enhancing the anchoring effect of the anchor cable [3]. Therefore, auxiliary pier are widely used in long-span cable-stayed bridges [4; 5; 6]. To give full play to the auxiliary pier, the positive (upward) reaction on the auxiliary pier needs to preserve a reliable magnitude to prevent the upward deflection of the girder during both the construction and operation stages [7]. The auxiliary pier bearing separation will change the structure's stress system and cause boundary nonlinearity, which has an adverse impact on the structure's stress state [1]. Therefore, not only the traditional criteria of \"the tower being vertical, the girder being horizontal and a uniform distribution of cable forces\" (C1-C3) but also the criterion of \"no negative reaction under the live load\" (C4) should be satisfied during the design state of a completed bridge [1]. The criterion requires that the bearing reactions of the transition piers and auxiliary piers should have a sufficient positive reaction under the dead load without a negative reaction under the live load. However, it is difficult to totally achieve the objective service stage (OSS) considering the unpredictable multisource construction uncertainties [8; 9], and the NRRAP will be further increased when considering traffic growth at the operation stage [10; 11]. Therefore, it is crucial to decide an economic and practical strategy against the NRRAP at the operation stage on the basis of ensuring \"the tower being vertical, the girder being horizontal and a uniform distribution of cable tensions\". For long-span asymmetric cable-stayed bridges, the NRRAP can be reduced by girder counterweight or cable tension optimization in the construction stage [12; 13]. However, the parallel strand cable can hardly be adjusted at the operation stage [14]. Because the strands are stretched one-by-one and anchored with wedge-type anchors, the strands at the tensioning end have to be cut according to the required tensioning length after cable tension adjustment. After the cable tension adjustment, the protective cover should be installed and filled with oil to protect the anchor head, which will further increase the difficulty of cable tension adjustment at the operation stage [15]. In addition, it is extremely difficult to construct a girder counterweight inside the main girder at the operation stage because of the restricted construction space, which is accompanied by cable tension optimization [16]. Due to the high cost of tensioning operations, bridges should be designed so that the stay cables are prestressed the minimum number of times and no final tuning retensioning operation is sought after construction, especially for cable-stayed bridges with parallel strand cables [17]. In summary, it is infeasible to construct the girder counterweight or optimize the cable tension at the operation stage. Furthermore, an optimal coping strategy that can successfully control the NRRAP at the operation stage has not been reported in the literature thus far. In addition to multiple control criteria, it is also necessary to quantify the various uncertainties by a probabilistic analysis at the construction and operation stages. Therefore, reliability-based design optimization (RBDO) is essential to decide an optimal coping strategy [7]. Massive simulations and analyses are required when coupling the RBDO and the multiple control criteria. However, finite element model (FEM) analysis is very time-consuming due to the strong nonlinearity of a cable-stayed bridge. Nested RBDO based on an FEM analysis would make it computationally expensive and even intractable.