# A Two-Stage Multi-Objective Optimization Algorithm for Solving Large-Scale Optimization Problems
## 3 Proposed Algorithm
### Pareto Optimal Subspace Learning
\begin{array}{rl}maximize&\frac{\sum_{i=1}^{m}f_{i}(x)s_{i}}{\sqrt{ \sum_{i=1}^{m}f_{i}^{2}(x)\sqrt{\sum_{i=1}^{m}s_{i}^{2}}}}\\ s.t.&x\in\Omega\\ \end{array} \right. \tag{3}\]

In short, we can find the nearest solution of the angle between the target vector and the corresponding axis. In addition, the lowest point and ideal point are obtained by extracting the maximum and minimum values from each dimension of the extreme point, and the mapping of each reference line is realized by line 14 in the algorithm. The PF in the multi-objective optimization problem may not occupy the entire target space, or only a part of N reference lines pass through the PF. In this case, the final solution will lose diversity. Therefore, the purpose of drawing reference lines is to keep the preset N reference lines within PF as far as possible so as to maintain the diversity of solutions. Finally, each solution is obtained by solving SOP (4) with a set of reference lines r, and the reference lines in r are evenly distributed in PF, and each solution in Q has the smallest cosine value for each reference line, so the solution obtained has good diversity. ```
Input: The Pareto-optimal subspace \(\Omega\), the updated reference lines r = \(\{r_{1},r_{2},\ldots,r_{n}\}\); Output: P
1.\(\text{\emph{extreme\_points}}\leftarrow\emptyset\)
2.fori \(\leftarrow\) 1 to m do
3.\(\text{s}=[s_{1},s_{2},\ldots,s_{m}]=[0,\ldots 1,\ldots,0]\leftarrow\) Initialize a vector of m elements with zeros except for the i-th element with one;
4.\(x^{*}\leftarrow\) Solve the single-objective optimization problem
5.\(\text{\emph{extreme\_points}}\leftarrow\text{\emph{extreme\_points}}\cup f (x^{*})\);
6.end
7.\(\text{\emph{nadir\_point}}\leftarrow\emptyset\)
8.\(\text{\emph{ideal\_point}}\leftarrow\emptyset\)
9.fori \(\leftarrow\) 1 to m do
```

**Algorithm 5** Diversity Maintaining

## 4 Experiments and Results

In order to verify the effectiveness of the proposed algorithm, we compare it with four other advanced algorithms, namely NSGA-III [3], MOEA/D [2], MOEA/DVA [22], and LMEA [16]. on some standard test suits, namely DTLZ [30], WFG [31], UF [32], and LSMOP [33]. All experiments in this paper were run in MATLAB R2020b on a personal computer. The Windows 10 64-bit system is configured with an Intel (R) Core (TM) i7 4720 HQ CPU and 12G of RAM. The computer is manufactured by ASUS in Suzhou, China. ### Experimental Setting

For a fair comparison, all comparison algorithms use the recommended parameter values to achieve the best performance. The parameters of the experiment are set as follows: All comparison algorithms simulate binary crossover and polynomial mutation operations to generate offspring. For each test problem, the crossover probability is \(\mathrm{p_{c}}=1.0\). Crossover probability is an important parameter in genetic algorithms used to control the probability of individuals performing crossover operations during the evolutionary process. The \(\mathrm{Pc}\) is a value between 0 and 1, representing the probability of two parent individuals performing crossover operations. And the mutation rate is \(\mathrm{p_{m}}=1/\mathrm{D}\), where \(\mathrm{D}\) is the number of decision variables. Mutation rate serves as a crucial parameter in algorithms, governing the probability of random alterations in individual genes. For all algorithms, the population size is set to 100. For test questions with decision variables of 100, 500, and 1000, the maximum evaluation times are set to 1,000,000, 5,000,000, and 15,000,000, respectively. Two widely used performance indicators, namely inverse generation distance (IGD) [34] and HV [35], are used to evaluate the performance of the compared algorithms. IGD is an index that can evaluate convergence and uniformity performance at the same time. The calculation method is as follows. \[\left\{ \begin{aligned} &\mathrm{IGD}(\mathrm{PS},\mathrm{PS}^{*})= \frac{\sqrt{\sum_{i=1}^{\mathrm{PS}}d_{i}}}{|\mathrm{PS}|}\\ & d_{i}=\min_{y\in\mathrm{PS}^{*}}\left\|F\left(x^{i}\right)-F(y) \right\|,\ \ x\in\mathrm{PS}\end{aligned} \right.