# A Two-Stage Multi-Objective Optimization Algorithm for Solving Large-Scale Optimization Problems
## 3 Proposed Algorithm
### Pareto Optimal Subspace Learning
#### 4.2.4 Efficiency Comparison Between LSMOEA-VT and Other Algorithms

In order to further study the performance of LSMOEA-VT, we compare the running time of five algorithms on some test problems, in which the number of decision variables of DTLZ1, DTLZ2, WFG3, UF9, and UF10 is 100, and the number of LSMOEA-VT is 300. The results are shown in Table 4. From the table, we can see that LSMOEA-VT spends the least time on DTLZ1, DTLZ 2, WFG3, UF9, and UF10 issues, and the time spent on the LMEA algorithm based on the grouping strategy is also much less than MOEA/DVA, because MOEA/DVA spends a lot of time on the analysis of decision variables, resulting in poor performance. At the same time, on the LSMOP test suite, because the program is relatively simple and ignores the impact of performance, NSGA-III takes the least time among all algorithms. The time required for LSMOEA-VT is longer than that for LMEA. This is because the computational cost for maintaining the population diversity in the LSMOEA-VT algorithm is higher than that of LMEA. Therefore, although the population diversity in the LSMOEA-VT algorithm is increased, it consumes more computational resources. Figure 4: IGD metric values of LSMOEA-VT on DTLZ1, DTLZ2, WFG3, UF9, and UF10 with different numbers of decision variables. \begin{table}
\begin{tabular}{c c c c} \hline
**Problem** & **Obj.** & **2000-Variable** & **4000-Variable** \\ \hline DTLZ1 & 5 & \(9.3525\times 10^{-1}\) (\(7.37\times 10^{-3}\)) & \(9.3571\times 10^{-1}\) (\(5.36\times 10^{-3}\)) \\ DTLZ2 & 5 & \(6.2454\times 10^{-1}\) (\(5.17\times 10^{-3}\)) & \(6.2331\times 10^{-1}\) (\(4.65\times 10^{-3}\)) \\ DTLZ3 & 5 & \(6.2466\times 10^{-1}\) (\(4.94\times 10^{-3}\)) & \(6.2370\times 10^{-1}\) (\(1.53\times 10^{-2}\)) \\ DTLZ4 & 5 & \(5.7154\times 10^{-1}\) (\(4.53\times 10^{-2}\)) & \(5.8147\times 10^{-1}\) (\(2.34\times 10^{-2}\)) \\ DTLZ5 & 5 & \(3.3281\times 10^{-2}\) (\(2.87\times 10^{-4}\)) & \(3.2918\times 10^{-2}\) (\(3.49\times 10^{-4}\)) \\ DTLZ6 & 5 & \(3.3110\times 10^{-2}\) (\(1.41\times 10^{-3}\)) & \(3.4250\times 10^{-2}\) (\(1.37\times 10^{-4}\)) \\ DTLZ7 & 5 & \(1.2547\times 10^{-1}\) (\(4.85\times 10^{-3}\)) & \(1.2963\times 10^{-1}\) (\(3.10\times 10^{-3}\)) \\ WFG3 & 5 & \(5.8532\times 10^{-1}\) (\(2.54\times 10^{-3}\)) & \(5.8791\times 10^{-1}\) (\(3.76\times 10^{-3}\)) \\ UF9 & 3 & \(7.0363\times 10^{-1}\) (\(1.49\times 10^{-2}\)) & \(7.1057\times 10^{-1}\) (\(1.22\times 10^{-2}\)) \\ UF10 & 3 & \(1.6887\times 10^{-1}\) (\(7.96\times 10^{-3}\)) & \(1.6449\times 10^{-1}\) (\(1.60\times 10^{-2}\)) \\ \hline \end{tabular}
\end{table} 
Table 3: HV metric values of LMEA on DTLZ1–DTLZ7, WFG3, UF9, and UF10, with 2000 and 4000 decision variables. ## 5 Conclusions and Future Work

In this paper, a two-stage large-scale optimization algorithm based on decision variable clustering (LSMOEA-VT) is proposed to solve large-scale optimization problems. The decision variables are classified in LSMOEA-VT, and the dimension reduction method is used to optimize the variables that affect the evolutionary convergence. Then, a non-dominated dynamic weight aggregation method is used to solve a set of single-objective optimization problems with reference lines through Pareto optimal subspace to enhance the diversity of the population. In the above experiments, we have carried out experimental comparisons on some low-dimensional benchmark functions, namely the UF test suite, and the experiments show that we can obtain satisfactory results with less computational cost. In addition, we also compared with other algorithms on DTLZ, WFG, and UF test suites when the decision variable is 100, 500, and 1000, and obtained satisfactory results in convergence and diversity, thus verifying the effectiveness and efficiency of our algorithm. Finally, we also try the problem of 4000 decision variables to verify the scalability of our algorithm. Although our algorithm has excellent performance in solving large-scale optimization problems, there are still some problems to be solved.