# 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
However, at any time \(t\), \(POF(t)\) is \(F_{2}=1-\sqrt{F_{1}}\) and did not change with time. Figure 3 shows the solution sets obtained by three algorithms for solving FDA1 at three different times in the tenth cycle. The corresponding environmental change factors in these three moments were \(\tau_{T}=10\), \(n_{T}=10\). Figure 4 describes the solution sets obtained by three algorithms, also for FDA1, under the parameters \(\tau_{T}=25\) and \(n_{T}=10\). From the two figures, it can be seen clearly that the solution sets obtained by dMOEA/D-\(L_{p}\) were the most uniformly convergent to \(POF(t)\) in different environments. The results of the other two algorithms in mapping significantly deviated from \(POF(t)\) and the distributions were uneven. Some components (\(x_{III}\)) of the Pareto optimal solutions in FDA2 varied sinusoidally with the function \(H(t)/4\), so the \(POS(t)\) of the FDA2 changed with time. Furthermore, the \(POF(t)\) of FDA2 also changed with time sinusoidally because \(f_{2}=1-(f_{1}/g)^{2^{H(t)}}\). Figure 5 shows the solution sets obtained by three algorithms for solving FDA2 at three different times in the tenth cycle. Environmental change factors were set to \(\tau_{T}=15\) and \(n_{T}=5\). It can be seen from Figure 5 that the Pareto front shape of the problem changed from convex to concave. The solutions sets obtained by dMOEA/D-\(L_{p}\) at three different times were closest to \(POF(t)\), and the convergence distribution of dMOEA/D-\(L_{p}\) was better than the other two algorithms. The \(POS(t)\) of the FDA3 also changed with time. Its \(POF(t)\) was \(f_{2}=(1+G(t))\times(1-\sqrt{f_{1}/(1+G(t))})]\), and also changed with \(G(t)\). Moreover, the density of the \(POF(t)\) also varied with time. Figure 6 shows the distributions of the solution sets obtained by three algorithms at three different times. From left to right, the corresponding environmental change amplitude of the problem in Figure 6 varied from small to large, and the distributions of \(POF(t)\) shifted from top to bottom. All three plots of Figure 6 show that the distribution and convergence of dMOEA/D-\(L_{p}\) were the best. Some components of FDA5's optimal variables (\(x_{II}\)) changed with \(G(t)\), so its \(POS(t)\) changed with time. When the number of objectives \(m\) was 3, the \(POF(t)\) of FDA5 was 1/8 of a sphere at any time, but the radius of the sphere varied between 1 and 2 with time \(t\). Figures 8-10 plot the solution sets obtained by three algorithms on FDA5 at three different times. The environmental change amplitude of the problem varied from large to small in the three times, and the spherical radius of the \(POF(t)\) grew from 1 to 2. Figures 8-10 show that dMOEA/D-\(L_{p}\) could best track the \(POF(t)\), and the convergence and distribution of the solution sets obtained by dMOEA/D-\(L_{p}\) were obviously superior to those obtained by the other two algorithms. Figure 8: Solution sets obtained by three algorithms at \(t=10.2\) on FDA5 in case of \(\tau_{T}=15\), \(n_{T}=5\). The \(POS(t)\) and \(POF(t)\) of FDA4 were similar to FDA5, except that the spherical radius of the \(POF(t)\) remained the same value (one), not varying with time \(t\). Similar comparing results happen for FDA4, which can be seen in Figure 7. It indicates that dMOEA/D-\(L_{p}\) performed with obvious superiority over the other two algorithms. ## 5 Conclusions

In order to handle dynamic multi-objective optimization problems well, a memory-enhanced dynamic multi-objective evolutionary algorithm based on \(L_{p}\) decomposition (denoted by dMOEA/D-\(L_{p}\)) is proposed in this paper. Specifically, dMOEA/D-\(L_{p}\) decomposes a dynamic multi-objective optimization problem into a number of dynamic scalar optimization subproblems and optimizes them simultaneously. The \(L_{p}\) decomposition method is referred and modified to use.