# A Two-Stage Multi-Objective Optimization Algorithm for Solving Large-Scale Optimization Problems
## 3 Proposed Algorithm
### Pareto Optimal Subspace Learning
\tag{4}\]

PS represents the Pareto optimal solution set in the population, and PS* represents the real Pareto optimal solution set. \(\mathrm{d_{i}}\) represents the minimum Euclidean distance between the candidate solution in PS and the real solution in PS* after mapping to the target space. The smaller the IGD value, the better the performance of the algorithm. HV evaluates the performance of the algorithm in convergence and diversity in a quantitative way bycalculating the sum of the super volume of the hypercube composed of all nondominated solutions and the nadir point in the population. \[ \begin{cases}\text{HV}\Big{(}\text{PS},x^{ref}\Big{)}=volume\begin{pmatrix} \parallel\text{PS}\parallel\\ \bigcup\limits_{i=1}^{|\text{PS}\parallel}c_{i}\\ d_{i}=\min\limits_{y\in\text{PS}^{i}}\left\lVert F\big{(}x^{i}\big{)}-F(y) \right\rVert,\;\;x\in\text{PS}\end{cases}  \tag{5}\]

\(\text{c}_{i}\) represents a hypercube formed by a non-dominated solution in PS and a reference point xref as a diagonal line. The calculation method is shown in Equation (6). The larger the HV value, the closer the solution of the algorithm is to the true PF, and the better the performance of the algorithm. All algorithms are independently run 20 times for each test problem to obtain statistical results. \[ \begin{cases}volume(c_{i})=\begin{cases}\prod_{k=1}^{m}\Big{(}x_{k}^{ref}-f_{k} \big{(}x^{i}\big{)}\Big{)},\;\forall k\Big{(}x_{k}^{ref}-f_{k}\big{(}x^{i} \big{)}>0\Big{)}\\ 0,\exists k\Big{(}x_{k}^{ref}-f_{k}\big{(}x^{i}\big{)}<0\Big{)}\end{cases}\end{cases}  \tag{6}\]

### Experiments and Analysis

#### 4.2.1 Results of Problems with 30 Variables

Figures 2 and 3 show the PF values obtained by the five algorithms on the UF2 and UF4 test functions. From the figure, we can clearly see that NSGA-III and MOEA/D have obtained satisfactory convergence results on the UF2 problem, but the final population results obtained by them do not cover the entire Pareto front. The convergence result of LMEA is very good, but the diversity of the population is not as good as LSMOEA-VT. On the UF4 problem, NSGA-III and MOEA/D algorithms perform poorly, and the population results do not fully converge to the real PF. The convergence and diversity of the LSMOEA-VT population have obtained good results in both UF2 and UF4 problems. Figure 2: The PF obtained by five algorithms on UF2 function with 30 decision variables. (**a**) NSGA-III; (**b**) MOEA/D; (**c**) MOEA/DVA; (**d**) LMEA; (**e**) LSMOEA-VT.

#### 4.2.2 Performance Comparison Between LMEA and Existing MOEAs on Large-Scale MaOPs

Table 1 shows the statistical results of IGD values obtained by 5 comparison algorithms through 20 independent runs on 8 test problems (100, 500, and 1000 decision variables). The best results of each test are shown in bold. Wilcoxon rank sum test is also used. The symbols \"+\", \"\(-\)\", and \"\(\approx\)\" indicate that the results are significantly better, significantly worse, and basically the same as LSMOEA-VT. Besides, the best results in Tables 1 and 2 are shown in bold. From the table, we can find that LSMOEA-VT achieves the best results for most test problems in terms of the statistical results of IGD. NSGA-III and MOEA/D can perform well in some test functions, but their overall performance is not as good as MOEA/DVA, which is used to solve large-scale multi-objective optimization problems, LMEA, and our LSMOEA-VT algorithm. On the other hand, we can see that when the number of decision variables increases from 100 to 1000, the IGD values obtained by LMEA and LSMOEA-VT on each test function are very good, which shows that LMEA and LSMOEA-VT have good expansibility. On the other hand, on the DTLZ1-DTLZ4 problem, LSMOEA-VT and other comparison algorithms have achieved different and better results, but on the DTLZ7 test problem, LSMOEA-VT performance is always higher than other comparison algorithms.