## 1 Introduction
### Related Works  
A robust design optimization aims to identify a robust point that is less prone to failure from parameter variations [8; 9]. First, the variability in design parameters is prescribed, either through measurement or assumption, and then represented using mathematical concepts such as permissible values and probability distributions. Second, it is assumed that the performance function is analytically known. This enables the calculation of a robustness measure for each design parameter, which is subsequently integrated into the performance function [10; 11; 12; 13]. Such analyses identify an optimal point, which is a trade-off between robustness and performance. This method is effective when both parameter variability and the explicit expression of the performance function are known. When parameter variability or the performance function is unknown, an alternative approach is required. The box-shaped solution space is an alternative for the unknown parameter variability. The box-shaped solution space is represented by intervals for each design parameter. The box-shaped solution space can decouple design parameters, making design intervals for one parameter independent of others. The volume of the box-shaped solution space measures tolerance to parameter variations, reflecting its robustness. In fact, box-shaped solution spaces have been widely used in engineering design. For instance, [PERSON]. [14] proposed a systematic design method for vibrating systems to identify the box-shaped solution space. Additionally, [PERSON]. [15] constructed a box-shaped solution space to identify non-intuitive robot designs via cascaded optimization. Several approaches have been proposed to identify the box-shaped solution space [16; 17; 18]. The stochastic method presented by [PERSON] and [PERSON] integrates querying with online learning [16]. It explores the candidate solution space through stochastic sampling and adjusts the boundaries based on the findings. This method first removes designs outside the complete solution space and then explores unprobed design spaces. Additionally, in [19], [PERSON] et al. investigated the efficiency of this stochastic method. However, using Monte Carlo sampling to estimate boundaries may prevent this method from achieving the maximum volume. In addition, in [18], [PERSON] et al. proposed a sample-based method that integrates metaheuristic algorithms with the DIRECT method. The metaheuristic algorithm maximizes the volume of the box-shaped solution space, while the DIRECT algorithm verifies the feasibility of the obtained solution space. Another approach to seeking the solution space was presented by [PERSON] et al., which combines a cellular evolutionary algorithm and an interval arithmetic method [20]. Nevertheless, the applicability of interval arithmetic is limited because its accuracy is problem-specific and cannot be generally evaluated [21]. It is observed that the mentioned approaches focus only on maximizing the volume of the solution space, overlooking its performance optimization. In [22], [PERSON] and [PERSON] further modified the approach in [20]. They transform the single-objective problem into a multi-objective one, generating a set of Pareto-optimal solution spaces based on conflicting performance functions. This allows the optimization of performance while ensuring robustness. Once the set of Pareto-optimal solutions is identified, engineering designers can select their preferred option. However, they assume that the performance function is explicitly defined, making it unsuitable for black-box performance functions. The above literature about robust design can be summarized from different aspects according to the (a) types of design; (b) assumptions imposed on the parameter variability; (c) assumptions imposed on the performance function; and (d) types of optimization problem.