# A Two-Stage Multi-Objective Optimization Algorithm for Solving Large-Scale Optimization Problems
## 3 Proposed Algorithm
### Pareto Optimal Subspace Learning
This is because the DTLZ7 problem has an irregular Pareto front, while MOEA/D and MOEA/DVA both deal with diversity problems based on decomposition, which is not applicable to the DTLZ7 problem. On the contrary, the strategy of maintaining population diversity proposed in algorithm 5 can well deal with this irregular Pareto front. Finally, the performance of MOEA/DVA is better than LSMOEA-VT in UF9 and UF10 testing. This is because in these problems, MOEA/DVA optimizes all variables related to convergence and diversity as diversity-related variables, while MOEA/DVA uses a uniform sampling method to initialize the diversity-related variables, which can obtain a good candidate solution distribution in the UF problem. Figure 3: The PF obtained by five algorithms on UF4 function with 30 decision variables. (**a**) NSGA-III; (**b**) MOEA/D; (**c**) MOEA/DVA; (**d**) LMEA; (**e**) LSMOEA-VT.

\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Problem** & **Obj. ** & **Dec.** & **NSGA-III** & **MOEA/D** & **MOEA/DVA** & **LMEA** & **LSMOEA-VT** \\ \hline \multirow{4}{*}{DTLZ1} & \multirow{4}{*}{5} & \multirow{4}{*}{500} & 4.2683 \(\times\) 10\({}^{+0}\) & 1.3858 \(\times\) 10\({}^{-1}\) & 7.6881 \(\times\) 10\({}^{-2}\) & **6.2269 \(\times\) 10\({}^{-2}\)** & 6.3955 \(\times\) 10\({}^{-2}\) \\  & & & (1.03 \(\times\) 10\({}^{+0}\)) \(-\) & (2.26 \(\times\) 10\({}^{-3}\)) \(-\) & (4.73 \(\times\) 10\({}^{-4}\)) \(-\) & (**3.87 \(\times\) 10\({}^{-4}\))** + & (5.62 \(\times\) 10\({}^{-4}\)) \\  & & & 7.3485 \(\times\) 10\({}^{+1}\) & 1.7756 \(\times\) 10\({}^{-1}\) & 7.8375 \(\times\) 10\({}^{-2}\) & 6.3324 \(\times\) 10\({}^{-2}\) & **6.3173 \(\times\) 10\({}^{-2}\)** \\  & & & (8.54 \(\times\) 10\({}^{+0}\)) \(-\) & (5.21 \(\times\) 10\({}^{-2}\)) \(-\) & (2.74 \(\times\) 10\({}^{-3}\)) \(-\) & (6.25 \(\times\) 10\({}^{-4}\)) \(-\) & (**6.17 \(\times\) 10\({}^{-4}\))** \\  & & & 2.2688 \(\times\) 10\({}^{+2}\) & 1.9744 \(\times\) 10\({}^{-1}\) & 8.0133 \(\times\) 10\({}^{-2}\) & **6.1723 \(\times\) 10\({}^{-2}\)** & 6.2387 \(\times\) 10\({}^{-2}\) \\  & & & (6.17 \(\times\) 10\({}^{+0}\)) \(-\) & (5.07 \(\times\) 10\({}^{-2}\)) \(-\) & (1.57 \(\times\) 10\({}^{-3}\)) \(-\) & (**5.18 \(\times\) 10\({}^{-4}\))** & (6.33 \(\times\) 10\({}^{-4}\)) \\ \hline \multirow{4}{*}{DTLZ2} & \multirow{4}{*}{5} & \multirow{4}{*}{100} & 3.5217 \(\times\) 10\({}^{-1}\) & 3.5226 \(\times\) 10\({}^{-1}\) & 2.7182 \(\times\) 10\({}^{-1}\) & 2.1687 \(\times\) 10\({}^{-1}\) & **2.0285 \(\times\) 10\({}^{-1}\)** \\  & & & (5.72 \(\times\) 10\({}^{-4}\)) \(-\) & (2.45 \(\times\) 10\({}^{-7}\)) \(-\) & (5.62 \(\times\) 10\({}^{-6}\)) \(-\) & (2.97 \(\times\) 10\({}^{-3}\)) \(-\) & (**3.15 \(\times\) 10\({}^{-3}\))** \\  & & &In order to further study the performance of LSMOEA-VT on large-scale optimization problems, we have performed more tests on the recently developed large-scale optimization test suite, namely the LSMOP test suite. The LSMOP test suite consists of nine test problems, LSMOP1-LSMOP9. These test problems have complex variable relations on the Pareto set, and the decision variables are mixed and separated. Therefore, the LSMOP test problems can better reflect the challenges faced by large-scale optimization problems. From Table 2, we can see that LSMOEA-VT achieves the best performance on LSMOP1, LSMOP2, LSMOP3, LSMOP5, and LSMOP9. For LSMOP8, the IGD obtained by LSMOEA-VT is slightly worse than that of LMEA. This is because LSMOP8 is a multi-mode problem with a mixture of partially separable and completely separable decision variables. Because LSMOEA-VT achieves the efficient solution of LSMOPs from the perspective of decision variable grouping, the effect of the grouping method may be affected in the face of multimodal problems and the problem of partial separability of decision variables, resulting in poor performance of the algorithm. But overall, LSMOEA-VT has achieved good performance in LSMOP testing problems.