## 3 Application and Results
### Case Study II: The BNH Problem  
For this case study, we apply the ALMO approach to solve the BNH problem, which is a well-known benchmark problem in multi-objective constrained optimization. The BNH problem was originally introduced by [PERSON] and [PERSON] [26] and involves two objective functions and two non-linear constraints. The mathematical formulation of the BNH problem is as follows:  
\[\text{Minimize}\;f_{1}(\mathbf{x})=4x_{1}^{2}+4x_{2}^{2}, \tag{16}\] \[\text{Minimize}\;f_{2}(\mathbf{x})=(x_{1}-5)^{2}+(x_{2}-5)^{2}, \tag{17}\]  
subject to the following two constraints:  
\[c_{1}(\mathbf{x})=(x_{1}-5)^{2}+x_{2}^{2}-25\leq 0, \tag{18}\] \[c_{2}(\mathbf{x})=-(x_{1}-8)^{2}-(x_{2}+3)^{2}+7.7\leq 0, \tag{19}\]  
Figure 8: Case study I: (**A**) percentage of infeasible solutions found in the optimal Pareto front; (**B**) overall savings in constraint evaluations in optimizations with different values of \(\phi\) and \(\beta\).  
Figure 9: Case study I: percentage of savings in constraint evaluations in different optimization intervals using ALMO with two different configurations.  
where the decision variables \(x_{1}\) and \(x_{2}\) are bounded as  
\[0 \leq x_{1}\leq 5, \tag{20}\] \[0 \leq x_{2}\leq 3. \tag{21}\]  
Figure 10 shows the solution for the BNH problem implemented using NSGA-II and the pymoo Python library to establish a baseline for comparing the benefits of the ALMO framework.  
Subsequently, the ALMO approach is applied with two configurations of \(\phi\) and \(\beta\). In the first configuration, \(\phi\) is set to 100 and \(\beta\) is set to 0.7, resulting in an IGD of 0.002. The second configuration involves setting \(\phi\) to 200 and \(\beta\) to 0.9, and the IGD remains at 0.002. Figure 11 illustrates all feasible Pareto fronts obtained for both configurations, which indicates that both settings produce optimal solutions with minimal IGD values. This consistent performance suggests that the BNH problem is well handled by the ALMO framework across different parameter settings.  
We then examine the percentage of infeasible solutions present in the final Pareto fronts for various values of \(\phi\) and \(\beta\), as shown in Figure 12A. Interestingly, for all the configurations, the percentage of infeasible solutions is 0%. This is indicative of the fact that the BNH problem's feasible space is relatively easier for the learner to grasp compared to other problems like the OSY (Section 3.1) and due to the fact that neither constraint defined in the problem renders any solution from the unconstrained Pareto-optimal front infeasible. Therefore, the constraints might not significantly increase the difficulty of solving this problem. The absence of infeasible solutions highlights the robustness of the ALMO in maintaining feasibility while optimizing the objectives.  
Figure 11: Case study II: Pareto-front solution found using ALMO and two different configurations.  
Figure 10: Case study II: Pareto-front solution found by NSGA-II without AL using _pymoo_ library.  
Furthermore, the ALMO approach exhibits substantial savings in constraint evaluations, as shown in Figure 12B. For all tested configurations, the overall savings in the constraint evaluations remain close to 89%. This indicates that the ALMO framework is particularly effective in reducing the computational cost for the BNH problem, where the feasible region is well-defined and can be efficiently exploited by the learner.  
Lastly, the constraint evaluation savings are analyzed across different optimization intervals, revealing that ALMO becomes increasingly efficient as the optimization progresses, as shown in Figure 13. In the first interval of the optimization process, ALMO achieves around 50% savings, which increases to approximately 99% as the optimization proceeds. This trend reflects the growing confidence of the active learner in identifying feasible and optimal regions of the search space as the evolutionary process unfolds.