# 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
The best performance is highlighted with bold font. rGD(\(t\)): dynamic inverse generation distance. performance out of these four algorithms. In terms of the GD(\(t\))-metric, the same situations occur in Table 5. The best and the mean metric values in Tables 4 and 5 are mainly focused on dMOEA/D-\(L_{p}\), indicating that dMOEA/D-\(L_{p}\) behaved the best in almost all the test instances. The minimal std values obtained by dMOEA/D-\(L_{p}\) presented the best stability of the algorithm. We can conclude that the \(L_{p}\) decomposition method is effective and has some advantages over TCH, WS, and PBI decomposition approaches for decomposition-based dynamic multi-objective optimization. In the \(L_{p}\) decomposition method, \(p\in(0,\infty]\), when \(p=1\), \(g^{L_{p}}\) becomes the weighted sum method, and when \(p=\infty\), \(g^{L_{p}}\) becomes the standard Tchebycheff function. Figure 1 draws the contour lines of the \(L_{p}\) function with different \(p\) values showing that the search ability of \(L_{p}\) function decreased with the increase of \(p\). All the \(g^{L_{p}}\) functions suffered from the geometry issue except for the Tchebycheff function. Especially, a \(g^{L_{p}}\) function cannot identify the whole Pareto optimal front of an MOP when the curvature of the Pareto optimal front is larger than the curvature of the contour line of the chosen \(g^{L_{p}}\) function. Since the curvature of the Tchebycheff function is \(\infty\), it is able to identify Pareto optimal solutions for any type of geometry. Overall, for a \(g^{L_{p}}\) scalar function, its search ability and its robustness on problem geometries is a trade-off. The higher the search ability, the lower the robustness. If we can estimate the curvature of the Pareto optimal front in advance, then we can easily select a suitable \(g^{L_{p}}\) function based on its search ability. Based on our trial experiments on \(p\) for the FDA series problem in Section 4.3, \(p\) was set to 2 in advance for dMOEA/D-\(L_{p}\) in all experiments, which is consistent with the observation that if the curvature of the Pareto optimal front is quadratic, the most suitable \(g^{L_{p}}\) function should be the one using \(p=2\)[36]. Alternatively, for a new problem or a problem having a complex geometry, the use of the Tchebycheff function is a good choice. Compare dMOEA/D-\(L_{p}\) with DNSGA-II [16] and QICCA [7]

In order to test the performance of the proposed dMOEA/D-\(L_{p}\), we tested dMOEA/D-\(L_{p}\) on each test instance, and compared the results obtained by dMOEA/D-\(L_{p}\) with those obtained by the other two state-of-the-art algorithms: DNSGA-II [16] and QICCA [7]. Referring to the literature with open source codes of NSGA-II [21] and MOEA/D [23], we realized the DNSGA-II, QICCA, and dMOEAD-\(L_{p}\) with C++ language programming. The final rGD(\(t\))-metric statistical results are shown in Table 6, and the final GD(\(t\))-metric statistical results are shown in Table 7. Each table includes the best, mean, and standard deviation (std) of the performance metric values, and the best performance of each metric value is marked in bold. For intuitive analysis of the performance of three algorithms, Figures 3-7 describe the approximation set to the Pareto front in different times. As shown in Tables 6 and 7, in terms of the rGD and GD-metric, dMOEA/D-\(L_{p}\) achieved the best performance in almost all of the test instances. Compared with DNSGA-II and QICCA, dMOEA/D-\(L_{p}\) was competitive in solving this kind of dynamic multi-objective optimization problem. Figure 3: Solution sets obtained by three algorithms at different times on FDA1 in case of \(\tau_{T}=10\), \(n_{T}=10\). dMOEA/D-\(L_{p}\): memory-enhanced dynamic multi-objective evolutionary algorithm based on \(L_{p}\) decomposition; DNSGA-II: dynamic non-dominated sorting genetic algorithm II; POF: Pareto optimal front; QICCA: immune clonal coevolutionary algorithm for dynamic multi-objective optimization.