# A Two-Stage Multi-Objective Optimization Algorithm for Solving Large-Scale Optimization Problems
## 3 Proposed Algorithm
### Pareto Optimal Subspace Learning
#### 4.2.3 Performance of LMEA on Large-Scale MaOPs with 2000 and 4000 Decision Variables

In the previous part, we proved the excellent performance of the proposed algorithm LSMOEA-VT when the number of decision variables is from 100 to 1000. In this part, we further studied the performance of LSMOEA-VT when there are more decision variables. Table 3 shows the performance of the proposed LSMOEA-VT algorithm on DTLZ1, DTLZ7, WFG3, UF9, and UF10 when the decision variables are 2000 and 4000, respectively. From the table, we can see that LSMOEA-VT has obtained similar HV indicators on each test problem. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Problem** & **NSGA-III** & **MOEA/D** & **MOEA/DVA** & **LMEA** & **LSMOEA-VT** \\ \hline LSMOP1 & \(2.3421\times 10^{-1}\) & \(3.6152\times 10^{-1}\) & \(1.9591\times 10^{-1}\) & \(1.8182\times 10^{-1}\) & \(\mathbf{1.7651\times 10^{-1}}\) \\  & \((3.12\times 10^{-3})-\) & \((2.57\times 10^{-2})-\) & \((6.74\times 10^{-3})-\) & \((8.99\times 10^{-3})-\) & \(\mathbf{(8.99\times 10^{-3})}\) \\ LSMOP2 & \(1.6871\times 10^{-1}\) & \(2.4541\times 10^{-1}\) & \(1.6212\times 10^{-1}\) & \(1.5644\times 10^{-1}\) & \(\mathbf{1.5537\times 10^{-1}}\) \\  & \((2.51\times 10^{-3})-\) & \((8.98\times 10^{-4})-\) & \((2.61\times 10^{-3})-\) & \((2.54\times 10^{-3})\approx\) & \(\mathbf{(1.45\times 10^{-3})}\) \\ LSMOP3 & \(4.7176\times 10^{-1}\) & \(8.0682\times 10^{-1}\) & \(7.1024\times 10^{-1}\) & \(6.1232\times 10^{-1}\) & \(3.8682\times 10^{-1}\) \\  & \((1.30\times 10^{-1})-\) & \((5.37\times 10^{-1})-\) & \((4.81\times 10^{-2})-\) & \((5.68\times 10^{-2})-\) & \(\mathbf{(4.12\times 10^{-2})}\) \\ LSMOP4 & \(2.2422\times 10^{-1}\) & \(3.6237\times 10^{-1}\) & \(\mathbf{1.9776\times 10^{-1}}\) & \(2.1585\times 10^{-1}\) & \(2.2274\times 10^{-1}\) \\  & \((9.23\times 10^{-3})-\) & \((5.44\times 10^{-3})-\) & \(\mathbf{(3.29\times 10^{-3})+}\) & \((2.44\times 10^{-3})+\) & \((1.28\times 10^{-3})\) \\ LSMOP5 & \(3.5893\times 10^{-1}\) & \(7.9303\times 10^{-1}\) & \(4.1494\times 10^{-1}\) & \(3.7932\times 10^{-1}\) & \(\mathbf{3.5402\times 10^{-1}}\) \\  & \((1.82\times 10^{-1})\approx\) & \((6.23\times 10^{-2})-\) & \((3.83\times 10^{-2})-\) & \((1.65\times 10^{-2})-\) & \(\mathbf{(3.97\times 10^{-2})}\) \\ LSMOP6 & \(1.4049\times 10^{+0}\) & \(1.6947\times 10^{+0}\) & \(2.7213\times 10^{-1}\) & \(2.5820\times 10^{+4}\) & \(1.6651\times 10^{+3}\) \\  & \((2.52\times 10^{-1})+\) & \((4.22\times 10^{-1})+\) & \((1.97\times 10^{-1})-\) & \((3.71\times 10^{+3})-\) & \((4.33\times 10^{+3})\) \\ LSMOP7 & \(3.0012\times 10^{+0}\) & \(9.9436\)+0 & \(9.8291\times 10^{-1}\) & \(\mathbf{1.7422\times 10^{-1}}\) & \(2.8983\times 10^{+0}\) \\  & \((3.31\times 10^{-1})-\) & \((1.15\times 10^{-1})-\) & \((2.81\times 10^{-2})-\) & \(\mathbf{(3.02\times 10^{-2})+}\) & \((7.58\times 10^{-1})\) \\ LSMOP8 & \(3.2336\times 10^{-1}\) & \(6.1604\times 10^{-1}\) & \(3.5637\times 10^{-1}\) & \(\mathbf{3.1677\times 10^{-1}}\) & \(3.9824\times 10^{-1}\) \\  & \((1.02\times 10^{-2})+\) & \((1.76\times 10^{-2})-\) & \((6.92\times 10^{-2})+\) & \(\mathbf{(5.54\times 10^{-3})+}\) & \(\mathbf{(4.66\times 10^{-3})}\) \\ LSMOP9 & \(2.5286\)+0 & \(1.3600\)+0 & \(6.7058\times 10^{-1}\) & \(6.5163\times 10^{-1}\) & \(\mathbf{6.3776\times 10^{-1}}\) \\  & \((9.52\times 10^{-2})-\) & \((4.32\times 10^{-1})-\) & \((1.31\times 10^{-1})-\) & \((7.54\times 10^{-3})-\) & \(\mathbf{(2.77\times 10^{-2})}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: IGD metric values of the five algorithms on LSMOP1-LSMOP9. For further observation, Figure 4 shows the IGD index values of LSMOEA-VT on DTLZ 1, DTLZ 2, WFG3, UF9, and UF10, and 20 independent operation comparisons are made when the number of decision variables is 100, 500, 1000, 2000, and 4000. From the figure, we can see that with the increase of the number of decision variables, the IGD index obtained by LSMOEA-VT on test problems has hardly decreased, which shows that the LSMOEA-VT algorithm proposed by us has good scalability.