## 4 Conclusions  
In this paper, we introduce the active learning-based multi-objective optimization (ALMO) framework, designed to accelerate constrained evolutionary algorithms like NSGA-II by significantly reducing the number of costly constraint evaluations required. The core of this approach is the integration of an active learner, which dynamically queries new data points only when uncertainty arises. The effectiveness of the ALMO approach is demonstrated through its application to six benchmark problems, where it achieves significant constraint evaluation savings. A key finding from this study is that the ALMO framework maintains high solution quality, as evidenced by the IGD values, while improving computational efficiency by minimizing unnecessary evaluations. The success of the method relies on tuning parameters such as the learner's confidence threshold, \(\beta\), and the number of estimators, \(\phi\), for the  
Figure 31: Case study VI: percentage of savings in constraint evaluations in different optimization intervals using ALMO and two different configurations. Figure 30: Case study VI: (**A**) percentage of infeasible solutions found in the optimal Pareto front; (**B**) overall savings in constraint evaluations in the optimization with different values of \(\phi\) and \(\beta\). Random Forest model. Values of \(\beta\) above 0.7 are generally found to result in fewer infeasible solutions and higher savings in the constraint evaluations. We found that while ALMO delivers significant computational savings, the extent of these savings can decrease when more complex constraints require higher \(\beta\) values to maintain solution feasibility. This trade-off between feasibility and savings is a potential limitation, as higher \(\beta\) values increase the learner's confidence threshold, potentially requiring more constraint evaluations. However, even in such cases, ALMO still reduces the overall computational burden compared to traditional methods, though the savings may be less pronounced for problems with more challenging constraint landscapes. In future work, the ALMO framework can be extended in several promising directions. One potential improvement includes the exploration of advanced machine learning techniques, such as ensemble methods or deep learning architectures, to enhance the accuracy and adaptability of the active learner. Additionally, refining uncertainty quantification techniques could improve the querying strategy, leading to more efficient optimization by minimizing unnecessary evaluations. Another promising direction is the application of the ALMO framework to more diverse and complex multi-objective problems, which could further validate its effectiveness and robustness across various domains. Finally, investigating the integration of ALMO with other optimization algorithms may also provide insights into improving its performance in different optimization scenarios. Conceptualization, K.S. ; Methodology, K.S. ; Software, K.S. ; Validation, K.S. ; Formal analysis, K.S. ; Investigation, K.S. ; Resources, K.S. ; Data curation, K.S. ; Writing--original draft, K.S. ; Writing--review & editing, R.K.K. ; Visualization, K.S. ; Supervision, R.K.K. All authors have read and agreed to the published version of the manuscript. This research received no external funding. Not applicable. The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author. The authors  
* [PERSON] et al. (2008) [PERSON]; [PERSON]; [PERSON]; [PERSON] Multidisciplinary design analysis and optimization of space vehicle structures. In Proceedings of the AIAA SCITECH 2024 Forum, Orlando, FL, USA, 8-12 January 2024. * [PERSON] et al. (2008) [PERSON]; [PERSON]; [PERSON]. Solving engineering optimization problems with the simple constrained particle swarm optimizer. _Informatica_**2008**, _32_, 319-326. * [PERSON] and [PERSON] (2010) [PERSON] [PERSON]; [PERSON] [PERSON] Multi-objective Optimization Using Surrogates.