# Transfer Learning Based on Clustering Difference for Dynamic Multi-Objective Optimization
## 4 Experiments  
### Test Problems and Performance Indicators  
Our experiment was divided into two parts: the first part demonstrated the convergence and distribution uniformity of TCD-DMOEA by comparing it with several popular dynamic multi-objective algorithms. In the second part, through the comparison with Tr-DMOEA [28], it was possible to observe a reduction in the running time, effectively reducing the possibility of negative transfer. The entire experiment was based on MATLAB R2019b, running in a Windows 10 Pro. The strategy proposed by the experimental setup was mainly used during the initialization stage of the algorithm. A suite of preprocessing and transfer learning techniques allowed us to obtain a high-quality population adapted to the current environment, and evolution to obtain the optimal solution after modification. Theoretically, the target domain generated by the clustering difference strategy is more similar to the source domain, and the initial population improved by the transfer learning method is more adaptable to the changing environment, so as to obtain a solution closer to the actual DPF. To verify the effectiveness of the method, this section used 20 test functions and two related metrics to measure the convergence and uniformity of the algorithm, while using the running time to evaluate the negative transfer possibility of the algorithm. The 20 test functions used in this section were from CEC 2018: (1) DF function [41], and (2) F function [42]. The DF Benchmark Suite contains 14 questions (DF1-DF14) and the F Benchmark Suite contains six questions (F5-F10). The DF function is a diverse and unbiased benchmark problem, covering various attributes that represent various real scenes, such as time-dependent PF/PS geometry, irregular PF shape, disconnection, knee, etc. F5-F8 in F function have nonlinear correlations among decision variables. The PSs of F5-F7 are 1-D curves, and the PSs of F8 are 2-D surfaces. In F9, the environment changes smoothly in most cases. Sometimes, Pareto sets jump from one region to another. In F10, the geometry of two consecutive PFs is completely different. In 20 test benchmarks, the time parameter \(t\) was used here and defined as \(t=(1/n_{t})\left\lfloor(\tau/\tau_{t})\right\rfloor\), where \(\tau\), \(n_{t}\), and \(\tau_{t}\) represented the maximum number of iterations, the severity of the change, and the frequency of the change, respectively, as described in Table 1. Different kinetic parameters were set for the experiment. We set different variation severities, frequencies of change, and numbers of iterations so that each function could iterate 20 times, and the entire population size was set to 200, meaning that 200 solutions could be generated during evolution. In addition, the k value for K-means was set to 5. This experiment chose RM-MEDA [43] as the SMOEA optimizer of TCD-DMOEA. In this study, the following metrics were used to evaluate the performance of different algorithms. (1) Inverted generational distance (IGD [44; 45; 46]): IGD evaluates the convergence and diversity of algorithms by measuring the proximity between the real Pareto frontier and the Pareto frontier obtained by the algorithm, and the definition of the IGD indicator is calculated by Formula (14), where \(d\) is calculated by Formula (15). \[IGD(PF^{*}_{t},PF_{t})=\frac{\sum_{v\in PF_{t}}{{}^{*}}d(v,PF_{t})}{|PF_{t}{{}^ {*}}|} \tag{14}\]  
\[d(v,P_{t})=\min_{u\in P_{t}}\sqrt{\sum_{j=1}^{m}\left(f_{j}^{v}-f_{j}^{u} \right)^{2}} \tag{15}\]  
where \(PF^{*}_{t}\) is the standard POF of the \(t\)-moment, it is the POF obtained by the \(t\)-moment algorithm, and \(d\) is the Euclidean distance between the individual \(v\) on the \(PF^{*}_{t}\) and the individual closest to \(v\) in \(PF_{t}\). It can be seen that the evaluation method of IGD is for each individual in the standard POF. \(PF_{t}\) is used to find the closest point to it in the POF.