## 3 Application and Results
### Case Study III: Then TNK Problem  
In this section, we apply the ALMO approach to the well-known TNK problem introduced by [PERSON] et al. [27]. The TNK problem is a constrained bi-objective optimization problem notable for its non-linear and nonconvex constraints, making it a challenging test case for multi-objective optimization algorithms. The TNK problem is formulated as follows:  
\[\text{Minimize}\ f_{1}(\mathbf{x}) = x_{1}, \tag{22}\] \[\text{Minimize}\ f_{2}(\mathbf{x}) = x_{2}, \tag{23}\]  
Figure 12: Case study II: (**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 13: Case study II: percentage of savings in constraint evaluations in different optimization intervals using ALMO with two different configurations.  
subject to the constraints  
\[c_{1}(\mathbf{x}) =-x_{1}^{2}-x_{2}^{2}+1+0.1\cos\left\{16\arctan\left(\frac{x_{1}}{x_{ 2}}\right)\right\}\leq 0, \tag{24}\] \[c_{2}(\mathbf{x}) =\left(x_{1}-0.5\right)^{2}+\left(x_{2}-0.5\right)^{2}-0.5\leq 0, \tag{25}\]  
with the variable bounds  
\[0 \leq x_{1}\leq\pi, \tag{26}\] \[0 \leq x_{2}\leq\pi. \tag{27}\]  
The TNK problem's nonconvex constraints pose a significant challenge for traditional optimization algorithms. The results obtained using the NSGA-II implementation are shown in Figure 14 to establish a baseline for comparing the benefits of ALMO framework.  
Next, the ALMO approach is applied using two different configurations of \(\phi\) and \(\beta\). In the first case, \(\phi\) is set to 100 and \(\beta\) is set to 0.7, yielding an IGD of 0.002. In the second configuration, \(\phi\) is increased to 200 and \(\beta\) is set to 0.9, which results in the IGD being 0.001. Figure 15 displays all feasible Pareto fronts for both configurations, demonstrating that higher values of \(\phi\) and \(\beta\) generally result in a better IGD performance.  
Figure 16A shows that for \(\beta=0.7\), approximately 15% of the candidate solutions are infeasible, which decreases significantly to around 1% for \(\beta=0.9\). This pattern suggests that higher values of \(\beta\) result in better constraint handling, reducing the number of infeasible solutions in the final Pareto front. Compared to the OSY problem (Section 3.1), the TNK problem results in more infeasible solutions, suggesting that it is relatively more difficult for the learner to identify feasible regions in the search space.  
Figure 14: Case study III: Pareto-front solution found by NSGA-II without AL using _pymoo_ library.  
Figure 15: Case study III: Pareto-front solution found using ALMO and two different configurations.  
The ALMO approach also demonstrates substantial savings in the number of constraint evaluations required to solve the TNK problem, as shown in Figure 16B. For \(\beta=0.7\), the approach achieves an impressive 86% saving in constraint evaluations, while for \(\beta=0.9\), the savings are around 82%. These savings highlight the efficiency of ALMO in handling optimization problems with fewer constraint violations. Also, it should be noted that 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 17 illustrates the percentage of constraint evaluations saved across different optimization intervals. During the initial 20% of the optimization process, ALMO achieves around 40% savings, which steadily increase to approximately 90% by the final interval. This increase in savings as the optimization proceeds indicates that the ALMO approach becomes more confident in its predictions over time, resulting in greater computational efficiency during the later stages of the optimization as the Pareto front calculation progresses.