## 3 Application and Results
### Case Study I: The OSY Problem  
In this case study, we apply the ALMO approach to the OSY problem, originally introduced by [PERSON] and [PERSON] [25]. The OSY problem is a widely used benchmark for multi-objective optimization, characterized by non-linear constraints and a relatively complex solution space. The problem is formulated as follows:  
\[\text{Minimize }f_{1}(\mathbf{x})=-\Big{(}25(x_{1}-2)^{2}+(x_{2}-2)^ {2}+(x_{3}-1)^{2}+(x_{4}-4)^{2}+(x_{5}-1)^{2}\Big{)},\] (5) \[\text{Minimize }f_{2}(\mathbf{x})=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4 }^{2}+x_{5}^{  
\[0 \leq x_{1},x_{2},x_{6}\leq 10, \tag{13}\] \[1 \leq x_{3},x_{5}\leq 5,\] (14) \[0 \leq x_{4}\leq 6. \tag{15}\]  
To benchmark the ALMO's performance, we apply the NSGA-II algorithm without active learning to establish a baseline for comparing the benefits of the ALMO framework, as shown in Figure 6.  
Subsequently, the ALMO approach is employed to solve the OSY problem, using two configurations of \(\phi\) and \(\beta\), as shown in all feasible Pareto-front solutions in Figure 7. In the first scenario, \(\phi\) is set to 100 and \(\beta\) is set to 0.7, which results in an IGD of 0.002. In the second configuration, \(\phi\) is increased to 200 while \(\beta\) is set to 0.9, and the IGD remains at 0.002, indicating that both settings provide optimal solutions with low IGD values. However, the configurations show different performances in constraint handling, as discussed below.  
We next analyze the percentage of infeasible solutions present in the final Pareto fronts across various configurations of \(\phi\) and \(\beta\), as shown in Figure 8A. For \(\beta=0.6\), the percentage of infeasible solutions remains greater than or equal to 5% for all the values of \(\phi\). When \(\beta=0.7\), the percentage of infeasible solutions is consistently less than 5% for \(\phi\geq 100\), while for \(\beta=0.9\), the number of infeasible solutions drops to around 1%. It can be seen that as the \(\beta\) value is increased, there are, overall, fewer infeasible solutions in the final Pareto front.  
The ALMO approach also demonstrates significant savings in constraint evaluations, as shown in Figure 8B. The savings decrease steadily with higher \(\beta\) values, starting at around 70% for \(\beta=0.6\), rising to 65% for \(\beta=0.7\), 60% for \(\beta=0.8\), and reaching approximately 45% for \(\beta=0.9\). However, due to the inherent randomness in the population generated by the algorithm, some variability in the results is expected, leading to minor fluctuations in the observed savings and percentage of infeasible solutions.  
Figure 6: Case Study I: Pareto-front solution found by NSGA-II without AL using _pymoo_ library.  
Figure 7: Case Study I: Pareto-front solution found using ALMO and two different configurations.  
Lastly, the savings in constraint evaluations are examined across different optimization intervals. As the optimization progresses, the savings increase, demonstrating that ALMO becomes more effective at reducing the number of evaluations as the algorithm becomes more confident in identifying feasible regions. Figure 9 shows the percentage savings over the optimization interval, with ALMO achieving around 20% savings in the early stages, which increases to about 60% and 75% as the optimization proceeds. This trend further confirms the ability of ALMO to enhance efficiency in multi-objective optimization, particularly in later stages when the algorithm's predictions are more reliable.