## 1 Introduction  
Multi-objective optimization (MOO) addresses problems where multiple conflicting objectives must be optimized simultaneously. Unlike single-objective optimization, where a single optimal solution is sought, MOO aims to find a set of Pareto-optimal solutions that represent trade-offs among the objectives. Traditional evolutionary algorithms (EAs), such as NSGA-II (Non-dominated Sorting Genetic Algorithm II), are popular for solving such problems due to their robustness and ability to handle complex, high-dimensional search spaces [1]. However, NSGA-II, while effective, does not incorporate machine learning techniques such as active learning, which can dynamically adapt the optimization process based on uncertainty in constraint evaluations. Unlike NSGA-II, our proposed ALMO method integrates active learning to focus computational resources on critical areas, reducing unnecessary evaluations and improving computational efficiency. MOO algorithms have been extensively applied in engineering for the design and optimization of complex structures. [PERSON] et al. [2] achieved the multi-objective optimization of laminated composite beam structures where the objective function was to minimize the weight of the whole laminated composite beam and maximize the natural frequency. [PERSON] et al. [3] presented multi-objective structural optimization with conflicting objective functions and constraints, such as the natural frequencies of vibration and the load factors concerning the global stability of the structure. In the field of aerospace engineering, MOO has been employed to optimize the aerodynamic performance and structural integrity of aircraft components. [PERSON] et al. [4] presented the multi-objective and multidisciplinary design optimization of supersonic fighter wings. [PERSON] et al. [5] described the multi-objective optimization of composite aerospace structures while optimizing both the cost and weight of the structure. [PERSON] et al. [6] conducted a multidisciplinary design analysis and optimization of space vehicle structures by integrating structural, thermal, and acoustic analysis to optimize the spacecraft's overall performance while satisfying multiple design constraints. These diverse applications underscore the significance of MOO in solving complex, real-world engineering problems where multiple, often conflicting, criteria must be considered simultaneously. However, the aforementioned traditional algorithms are often computationally intensive, particularly when faced with complex constraints that necessitate extensive evaluations. The computational expense of MOO arises primarily from the need to evaluate the objective and constraint functions repeatedly across numerous candidate solutions. This is especially challenging in engineering and scientific applications when complex constraints require evaluations that are both time-consuming and resource-demanding. In many engineering applications, such as structural, aerospace, and civil engineering, these evaluations often involve expensive simulations or detailed analyses. One such method commonly used in these fields is the Finite Element Method (FEM), which is computationally expensive. The FEM is widely used in structural optimization to analyze stress, deformation, and other factors under various load conditions, and optimization using FEM can take hours or even days depending on the complexity of the model [7]. In aerospace engineering, MOO frequently deals with aerodynamic shape optimization when the evaluations involve high-fidelity Computational Fluid Dynamics (CFD) simulations. These simulations are also computationally heavy. Furthermore, thermal analysis, required for optimizing the thermal properties of aerospace materials and systems, also significantly adds to the computational cost [8]. To mitigate these challenges, surrogate models have been employed to approximate these expensive evaluations, thereby reducing the overall computational burden [9].