## 2 Optimization Model for the Embedded Retaining Wall
### Objective Function and Geotechnical Constraints of the Optimization Model  
The objective function aims to minimize the total length of the embedded retaining wall required to support the specified load. It includes three key parameters: \(L\), the total length of the wall; \(H_{nom}\), the excavation depth; and \(d_{nom}\), the embedment depth, as defined in Equation (1). \[\text{min}L=H_{nom}+d_{nom} \tag{1}\]  
Figure 1: Input parameters for the embedded retaining wall optimization model. Geotechnical analysis forms the basis for the optimization model. It considers all design constraints that ensure stability while minimizing the length of the wall. This analysis follows the guidelines of Eurocode 7 and considers both the bearing soil and all additional surcharge loads. The design must fulfill three main conditions:  
* Balance the horizontal forces from active and passive earth pressure. * Ensure that the moments generated by the active and passive earth pressure are balanced. * Prevent failure due to hydraulic heave. The proposed optimization model (see, Figure 1) includes the following input data (constants): The shear angle of the soil on the active and passive side \(\varphi^{\prime}_{k}\) [\({}^{\circ}\)], uniformly distributed surcharge load \(q_{Qk}\) [kN], slope angle \(\beta\) [\({}^{\circ}\)], whether groundwater is found behind the retaining wall at the surface level and in the excavated hole also at the surface level (see Figure 1) with the unit weight \(\gamma_{w}\) [kN/m\({}^{3}\)], cohesion of soil on the active and passive side \(C^{\prime}_{k}\) [kPa], reduction coefficient on the active side \(k_{u}\) [-] and reduction coefficient on the passive side \(k_{p}\) [-], unit weight of soil \(\gamma_{k}\) [kN/m\({}^{3}\)] and the height of the excavation \(H_{nom}\). To optimize the design based on Eurocode 7, for design approach 2, the following partial safety factors were used: the partial safety factor for permanent actions \(SF_{G}\) [-], the partial safety factor for favorable permanent actions \(SF_{G,fin}\) [-], the partial safety factor for variable actions \(SF_{Q}\) [-], the partial safety factor for the shear angle \(SF_{\varphi}\) [-], the partial safety factor for cohesion \(SF_{c}\) [-] and the partial safety factor for earth resistance \(SF_{Re}\) [-]. Furthermore, according to Eurocode standards, the design values for the groundwater pressure can be determined either by applying partial safety factors to the characteristic water pressures or by including a safety margin on the characteristic water level height. In this study, the latter approach was used, which assumes that the groundwater level reaches the ground surface and is at its highest possible point [25]. The optimum design is found based on the above-mentioned input data. In addition, the geotechnical conditions that must be met are defined using the equations summarized in Table 1. The optimization model ORW is therefore constrained by Equation (2), which limits the horizontal forces on the active side \(H_{Ed}\) with the resistance forces on the passive side \(H_{Rd}\). The total horizontal force \(H_{Ed}\) (Equation (2a)) on the active side of the embedded retaining wall is the sum of the active earth pressure and surcharge load \(H_{Ed,z,j}\) and the horizontal force due to the presence of groundwater \(H_{Ed,z,j}\). The horizontal forces due to soil and surcharge load are calculated using Equations (2b)-(2e). The horizontal forces due to the presence of groundwater are considered using Equations (2f)-(2i). The horizontal effective stresses \(\sigma^{\prime}_{a,d,i}\) required for the calculation of the horizontal forces and are calculated using Equations (2j)-(2m) and represent the design values as they are multiplied with partial safety factors. In addition, the effective vertical stresses \(\sigma^{\prime}_{v,k,i}\) caused by the weight of the soil are determined using Equations (2n)-(2r).