# A Two-Stage Multi-Objective Optimization Algorithm for Solving Large-Scale Optimization Problems
## 3 Proposed Algorithm
### Clustering of Decision Variables  
Algorithm 2 delineates the methodology for clustering decision variables. Consider a bi-objective optimization problem wherein each individual comprises five decision variables, denoted as \(\mathrm{x}_{1}\), \(\mathrm{x}_{2}\), \(\mathrm{x}_{3}\), \(\mathrm{x}_{4}\), and \(\mathrm{x}_{5}\). Firstly, two candidate solutions are randomly selected from the entire solution set. Subsequently, the variables of these selected solutions are perturbed ten times. The perturbed solutions are then normalized to form a fitting line, denoted as \(\mathrm{L}\). The equation \(\mathrm{f}_{1}+\cdots+\mathrm{f}_{\mathrm{m}}=1\) represents the direction of convergence. In high-dimensional decision spaces, dynamic clustering during optimization would introduce significant computational overhead, especially when evaluating complex objective functions. Offline clustering allows us to preprocess the decision space and reduce computational complexity during the optimization process; clustering provided a stable and consistent framework for interaction analysis. Dynamic clustering, while potentially more adaptive, could introduce variability in the sensitivity analysis due to changes in cluster boundaries during optimization. By performing clustering offline, we establish a fixed reference framework that facilitates clearer interpretation of the relationships between decision variables and objective functions. Following this, the angle between the fittingline L and the hyperplane normal is calculated. Variables exhibiting larger angles contribute more significantly to diversity, whereas those with smaller angles contribute more to convergence. It is noteworthy that the greater the number of candidate solutions selected for decision variables, the more angles associated with each variable will be generated. Therefore, to achieve more precise results, it is advisable to select as many solutions as feasible. Finally, the k-means clustering method is employed to partition all variables in the decision space into two groups. Variables with smaller average angles are categorized as convergence-related variables, while the remaining variables are classified as diversity-related variables.  
```
Input: pop, number of candidate solutionsSelNum, number of perturbationPerNum Output: [Diver,Conver]
1.for i = 1:n do
2. C \(\leftarrow\) Choose SelNum solutions from pop randomly;
3.for j = 1:SelNum do
4. Make perturbation for the i-th variable of C[j] for PerNum times to bring a generation SP and then normalize it;
5. Generate a line \(L\) for SP in objective space;
6. Angle[i][j] \(\leftarrow\) the angle between L and normal line of hyperplane;
7. MSE[i][j] \(\leftarrow\) the mean square error of the fitting;
8.endfor
9.endfor
10. CV \(\leftarrow\) {mean(MSE[i]) < \(1\times 10^{-2}\)};
11. [C\({}_{1}\), C\({}_{2}\)] \(\leftarrow\) apply k-means to group all variables into two categories adopting Angle as feature;
12.if C\({}_{\text{V}}\bigcap\)C\({}_{1}\
eq\emptyset\) and C\({}_{\text{V}}\bigcap\)C\({}_{2}\
eq\emptyset\) then
13. C\({}_{\text{V}}\leftarrow\) C\({}_{\text{V}}\bigcap\)C, C is either C\({}_{1}\) or C\({}_{2}\) depending on which of the average of Angle is smaller;
14.endif
15. DV \(\leftarrow\) {j \(\
otin\) CV }
```  
**Algorithm 2** Clustering of Decision Variables