## 5 Experiment Results  
Tables 4 and 5 present the values of various performance metrics achieved by different DMOEAs in solving 14 DMOPs of the CEC 2018 benchmark suite. As shown in the table, we can see that our proposed DMOEA (LR) achieves the best MGD in six DMOPs, the best MHOPs, and the best MACD in seven DMOPs, and these three numbers are larger than other DMOEAs. This confirms the superior performance of our proposed method for solving DMOPs. The main advantage of our proposed DMOEA is twofold. First, our proposed DMOEA considers predicting the positions of multiple key points instead of only the center point for \(PF\). This can adapt to more generalized \(PF\) shape changes. Secondly, we try to learn the relationship among different key points of \(PF\) by LR, instead of only exploiting the correlation within one signal point (e.g., the center point). Thus, our proposed method achieves the best overall performance. There are mainly two situations in which our proposed method achieves a relatively poor performance. The first one is when \(PF\) has almost a parallel motion overall, such as DF1. In such a case, DMOEAs that predict the \(PF\) based on only its center point may  
\begin{table}
\begin{tabular}{l l l} \hline \hline
**Problem** & **Objectives** & **PS** & **PF** \\ \hline \multirow{7}{*}{DF12} & \multirow{7}{*}{\(\min \begin{cases}f_{1}(x)=g(x)\cos(0.5\pi x_{1})\\ \cos(0.5\pi x_{2})\\ f_{2}(x)=g(x)\cos(0.5\pi x_{1})\\ \sin(0.5\pi x_{2})\\ f_{3}(x)=g(x)\sin(0.5\pi x_{1})\\ \end{cases} \)} & \multirow{7}{*}{\(0\leq x_{i=1,2}\leq 1\)} \\  & & \(\prod\limits_{j=1}^{2}mod(|\lfloor k_{t}(2x_{j}-r)\rfloor|,2)=0\) & \(\sum\limits_{i=1}^{3}f_{i}=1\) \\  & & \(x_{i}=\sin(tx_{1}),i=3,\ldots,n\) & \(0\leq f_{i=1,3}\leq 1\) \\ \cline{1-1}  & & \(k_{t}=\lfloor 10\sin(\pi t)\rfloor\) \\ \cline{1-1}  & & \(r=1-mod(k_{t},2)\) \\ \hline \multirow{7}{*}{DF13} & \multirow{7}{*}{\(\min\begin{cases}f_{1}(x)=g(x)\cos^{2}(0.5\pi x_{1})\\ f_{2}(x)=g(x)\cos^{2}(0.5\pi x_{2})\\ f_{3}(x)=g(x)\sum\limits_{j=1}^{2}[\sin^{2}(0  
achieve better performance than our proposed method that employs the correlation of multiple key points. This is because correlation learning introduces more complex patterns of environmental changes, which is helpful for DMOPs with irregular changes, but may not accurately capture simple rules. Another situation where our proposed method does not achieve the best performance is when the environment changes are too complex to predict. In this case, no prediction method achieves a good performance, while the simplest method may achieve the best performance. For example, FL achieves the best MIGD for DF11 and the best MHVD for DF14. Next, we illustrate the stability of our proposed method based on the box plots of performance metric values achieved by various algorithms. To avoid overly lengthy content, we took the MIGD metric and DF1-DF3 as case studies, as shown in Figure 2, for the following two reasons. First, the results are similar between different performance metrics. Second, these three cases include three scenes where the ranks of our algorithms are low, the best, and high, respectively. As we can see from Figure 2, the interquartile range achieved by our algorithm has a similar rank to the average value in each case. Additionally, the box plots achieved by our algorithm have no outliers. Therefore, our algorithm has a stable performance in solving DMOPs. Figure 2: The box plots of MIGD metric values achieved by various algorithms in solving DF1–DF3. (a) DF1. (b) DF2. (c) DF3.