## 2 Methodology
### Active Learning for Multi-Objective Optimization (ALMO)
```
Input: Population size (\(\mu\)), Offspring size (\(\lambda\)), Max generations (\(N\)), Confidence threshold (\(\beta\)), Re-training interval (\(m\))  Initialize population \(p\) with size \(\mu\)  Initialize infeasible_counter = 0 for each individual in \(p\)do  Evaluate true objective functions and constraints endfor for generation \(=1\) to \(0.1\cdot N\)do  Apply NSGA-II (generate offspring, evaluate objectives and constraints, perform selection) endfor  Train active learner (Random Forest Classifier) using data from the first 10% of generations for generation \(=0.1\cdot N+1\) to \(N\)do  Apply NSGA-II to generate offspring for each offspring solution do  Evaluate true objective functions for the offspring  Predict feasibility using the active learner if confidence score \(>\beta\)then  Accept/discard offspring based on prediction alone else  Perform true constraint evaluation for the offspring  Accept/discard offspring based on the true constraint evaluation  Add newly evaluated solution (feasible or infeasible) to the training set  Increment infeasible_counter by 1 endif endfor if infeasible_counter \(>m\)then  Re-train the active learner with updated data  Reset infeasible_counter to 0 endif  Select top \(\mu\) individuals based on NSGA-II's non-dominated sorting and crowding distance endfor  Evaluate the final Pareto-front solutions using true constraint evaluations  Discard any infeasible solutions from the final Pareto-front Output: Final Pareto-optimal solutions
```  
**Algorithm 2** ALMO: active learning for multi-objective optimization  
This ALMO framework significantly reduces the number of expensive constraint evaluations, particularly in the later stages of the optimization process when the active learner has become more confident in comparison to at the beginning of the optimization. By training the learner after the initial 10% of generations have been evaluated, we ensure that it has enough data to make accurate predictions while maintaining computational efficiency. In experiments with benchmark problems, we observed more than 50% savings in constraint evaluations, depending on the problem's complexity, the values of \(\beta\), and the number of estimators (\(\phi\)) in the Random Forest classifier. The experimental setup also revealed that the ALMO approach retains a high degree of accuracy, as measured by Inverted Generational Distance (IGD), while achieving significant reductions in the total number of function evaluations compared to the total number of evaluations required to conduct the optimization without the use of active learning. For comparison purposes, the solutions obtained using NSGA-II without the use of active learning were employed as the true Pareto front. The IGD measures the proximity of the approximated Pareto front to the true Pareto front. It is calculated as the average Euclidean distance from each point on the true Pareto front to its nearest point on the approximated front:  
\[IGD(Q)=\frac{1}{|Q^{*}|}\sum_{q^{*}\in Q^{*}}\min_{q\in Q}d(q^{*},q) \tag{4}\]  
where \(Q^{*}\) is the reference Pareto front, \(Q\) is the approximated front, and \(d(q^{*},q)\) is the Euclidean distance. _Low IGD values indicate that the solutions generated closely match the reference optimal solutions' objective functions._ Since optimal objective function values are achieved from corresponding optimal design variable values, it follows that the design variables are also close to the reference results. Thus, low IGD values provide a strong indication that both the objective and design variable values are comparable.