# Transfer Learning Based on Clustering Difference for Dynamic Multi-Objective Optimization
## 1 Introduction  
Dynamic multi-objective optimization problems (DMOPs) [1; 2] are optimization problems in which the objective function and decision variables are related to time (environment). Their optimal solution is a set of Pareto optimal solutions that change dynamically with time (environment). Different from solving the static multi-objective optimization problem, when dealing with this kind of optimization problem, it is necessary to not only optimize several conflicting objectives, but also deal with the changes of objective function and constraints at the same time. DMOPs are widely used to solve many real-world problems, and common application domains include scheduling [3; 4], control [5], chemistry [6], industry [7] and energy design [8]. An ideal dynamic multi-objective optimization algorithm should contain three necessary parts, namely change detection, change response and multi-objective optimization algorithm (MOEA) [9; 10; 11; 12]. When the time variable changes, the algorithm needs to detect the change of the objective function in time and respond to the change according to different types of changes, deal with the optimization problem after the change, and use the multi-objective optimization algorithm to iterate the population, then quickly find DPF and DPS at the current moment. In fact, DMOPs can be regarded as a static multi-objective optimization problem (SMOEA) under a set of discrete time variables that can convert complex dynamic characteristics into static processing, which is more convenient and easier to handle, but its disadvantages are also obvious. That is, the processing speed is slow, and timeliness cannot be guaranteed. When encountering an environment with a high frequency of changes, it often fails to achieve the purpose of quickly tracking the Pareto front,resulting in poor performance of the algorithm. Therefore, for ideal DMOEAs, a suitable change response mechanism is the top priority in dealing with DMOPs. The existing environmental response strategies can be divided into the following four categories: diversity maintenance, memory strategy, prediction mechanism and transfer learning [13; 14]. Diversity maintenance improves algorithm performance mainly by maintaining or improving population diversity. The specific process is shown in Figure 1a. [PERSON] et al. [15] used the perturbation method to divide the decision variables into time-dependent and time-independent according to the dependence of the decision variables on the time parameters, and respectively adopted the optimal solution of the corresponding decision variables of the co-evolution of two subpopulations. In addition, [PERSON] et al. [16] maintained population diversity by simulating magnetic particles, and then quickly converged to the Pareto front in the current environment. Generally speaking, the diversity maintenance strategy directly adopts the Pareto optimal solution set of the optimization problem in the previous environment as the initial population in the current environment. [PERSON] et al. [17] used an additional auxiliary strategy to maintain diversity that maintains two archives focusing on convergence and diversity, respectively. In addition, for some problems with biased characteristics, an interval mapping strategy is designed to make their solutions have good diversity. Based on this, [PERSON] et al. [18] divided the decision variables into three parts and adopted the methods of diversity maintenance, prediction and diversity introduction, respectively, to generate high-quality offspring individuals and speed up the convergence of the population. Diversity preservation methods have better performance for DMOPs with weaker changes, but when the optimal solution of the historical environment deviates far from the real Pareto front in the current environment, it will lead to poor problem tracking performance. Figure 1: Description of environmental response strategies and methods. (**a**) diversity maintenance. (**b**) Memory based approach.