# Memory-Enhanced Dynamic Multi-Objective Evolutionary Algorithm Based on \(L_{p}\) Decomposition
## 3 Memory-Enhanced Dynamic Multi-Objective Evolutionary Algorithm Based on \(L_{p}\) Decomposition
### Detailed Description of dMOEA/D-\(L_{p}\) and Its Time Complexity Analysis
For ease of expressions and comparisons, we call the version of dMOEA/D with the Tchebycheff decomposition approach as dMOEA/D-TCH, the version of dMOEA/D with the weighted sum decomposition approach as dMOEA/D-WS, the version of dMOEA/D with the PBI decomposition approach as dMOEA/D-PBI, and the version of dMOEA/D that uses \(L_{p}\) decomposition approach as dMOEA/D-\(L_{p}\). There are two main sets of experimental studies in this section: (1) We compare dMOEA/D-\(L_{p}\) with dMOEA/D-TCH, dMOEA/D-WS, and dMOEA/D-PBI to test the effectiveness of the \(L_{p}\) decomposition method for dynamic multi-objective optimization; (2) We compare dMOEA/D-\(L_{p}\) with two other state-of-the-art algorithms (DNSGA-II [23], QICCA [7]) to test the overall performance of the algorithm. ### Test Problems

We used FDA benchmark test instances [3,16] and their improved versions [30,32] as our test problems. Detailed definitions of these test problems are shown in Table 1. When the intensity of environmental change (\(n_{T}\)) is relatively small, the environmental change is less intense, and the number of \(\text{POF}(t)\) is less in each cycle. When the \(n_{T}\) value is large, the environmental change is more intense, and there will be many different \(\text{POF}(t)\) in each cycle. \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Problems** & **Objective Functions** & **Variable Bounds** & \(n\) \\ \hline FDA1 & \(\left\{\begin{array}{l}f_{1}(x_{1})=x_{1},f_{2}(x)=g\cdot h\\ g(x_{II})=1+\sum\limits_{x\in X_{1}}x_{2}^{2},h(x_{II}-G(t))^{2},h(f_{1},g)=1- \sqrt{f_{1}}/g\\ G(t)=\sin(0.5\pi t),t=\left\lfloor\tau/\tau_{\text{T}}\right\rfloor/n_{T}\end{array}\right.\) & \(x_{1}=(x_{1})\in[0,1]\) \\ \hline FDA2 & \(\left\{\begin{array}{l}f_{1}(x_{1})=x_{1},f_{2}(x)=g\cdot h\\ g(x_{II})=1+\sum\limits_{x\in X_{1}}x_{2}^{2},h(x_{II}f_{2},g)=1-\left(f_{1}/g \right)^{2}\left(x_{2(n_{1}+n_{2}),g}^{2}\right)\\ H(t)=2\sin(0.5\pi t-1),t=\left\lfloor\tau/\tau_{\text{T}}\right\rfloor/n_{T} \end{array}\right.\) & \(x_{1}=(x_{1})\in[0,1]\) \\ \hline FDA3 & \(\left\{\begin{array}{l}f_{1}(x_{1})=x_{1}^{T(t)},f_{2}(x)=g\cdot h\\ g(x_{II})=1+G(t)+\sum\limits_{x\in X_{1}}x_{2}^{2}\,(x_{II}-G(t))^{2},h(f_{1},g )=1-\sqrt{f_{1}}/g\\ G(t)=\left\lvert\sin(0.5\pi t)\right\rvert,F(t)=\left\lfloor\tau/\tau_{\text{T }}\right\rfloor/n_{T}\end{array}\right.\) & \(x_{1}=(x_{1})\in[0,1]\) \\ \hline FDA4 & \(\left\{\begin{array}{l}f_{1}(x)=(1+g(x_{II}))^{2}-11\cos(0.5\pi tx_{i})\\ f_{1}(x)=(1+g(x_{II}))\sin(0.5\pi tx_{i})\\ f_{m}(x)=(1+g(x_{II}))\sin(0.5\pi x_{1})\\ g(x_{II})=\sum\limits_{x\in X_{1}}(x_{i}-G(t))^{2},G(t)=\left\lvert\sin(0.5\pi t )\right\rvert,t=\left\lfloor\tau/\tau_{\text{T}}\right\rfloor/n_{T}\end{array}\right.\) & \(x_{1}=(x_{1})\in[0,1]\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Test problems used in our experiments. ### Performance Metric

The dynamic inverse generation distance rGD(\(t\)) [33] and dynamic generation distance GD(\(t\)) [34; 35] are used as the performance evaluation metrics of the DEMOAs, and are defined as follows, respectively:

\[\text{rGD}(t)=\frac{\sum_{j=1}^{|POF^{*}(t)|}d_{i}}{|POF^{*}(t)|},d_{i}=\min_{k=1 }^{|Q(t)|}\sqrt{\sum_{j=1}^{m}\left(f_{j}^{*(i)}-f_{j}^{(k)}\right)^{2}}, \tag{8}\]

\[\text{GD}(t)=\frac{\sqrt{\sum_{i=1}^{|Q(t)|}\left(d_{i}^{\prime}\right)^{2}}}{ |Q(t)|},d_{i}^{\prime}=\min_{k=1}^{|POF^{*}(t)|}\sqrt{\sum_{j=1}^{m}\left(f_{ j}^{(i)}-f_{j}^{*(k)}\right)^{2}}, \tag{9}\]

where \(POF^{*}(t)\) is a set of uniformly distributed points in the objective space along the \(POF\) at time \(t\), \(Q(t)\) is an approximation set to the \(POF\) obtained by the algorithm at the same time \(t\), \(f_{j}^{*(i)}\) is the \(j\)-th objective function value of the \(i\)-th sampling point in \(POF^{*}(t)\), and \(f_{j}^{(i)}\) is the \(j\)-th objective function value of the \(i\)-th point in \(Q(t)\). Moreover, \(d_{i}\) is the minimum Euclidean distance between point \(i\) of \(POF^{*}(t)\) and the points in \(Q(t)\).