# Research on Multi-Objective Evolutionary Algorithms Based on Large-Scale Decision Variable Analysis
## 2 Related Work
### Large-Scale Multi-Objective Evolutionary Algorithms  
Currently, several scalability improvement strategies have been developed to address large-scale multi-objective optimization problems (LSMOPs). The main strategies are as follows:  
The first strategy is the cooperative co-evolution (CC) framework, whose core idea is to group complex decision variables and optimize each group independently using different evolutionary algorithms, ultimately integrating the results to achieve overall optimization. [PERSON] and [PERSON] [36] first proposed the CC framework and, as algorithm research progressed, problem decomposition strategies evolved from single-dimensional decomposition [37] to bi-sectional decomposition [38]. Single-dimensional decomposition divides N-dimensional variables into N one-dimensional groups, while bi-sectional decomposition splits them into two N/2-dimensional groups. Although these strategies simplify the problem, they do not fully consider the correlation between variables. In this context, the GDE3 algorithm [39], which is based on differential evolution, employs a cooperative co-evolution approach to enhance convergence and diversity. Meanwhile, the AMOEBA algorithm [40], which mimics the evolutionary behavior of single-celled organisms, explores the solution space in a collaborative manner within multi-objective optimization. To overcome this limitation, researchers proposed various improved grouping methods, such as the random grouping method [41], which increases the possibility of grouping related variables randomly. Therefore, when applying the CC framework to LSMOPs, further exploration of other effective auxiliary strategies is needed to ensure the accuracy of grouping and optimization effectiveness. The second strategy is problem transformation, which involves reducing the dimensionality of the problem through transformation functions. For instance, the WOF [42] framework randomly fixes groups of decision variables and then optimizes the weight vectors of each group using an evolutionary algorithm. The LSMOF framework proposed by [PERSON] et al. [43] reconfigures the original problem into a low-dimensional single-objective optimization problem, guiding population evolution through direction vectors and weight variables. However, the distribution of good solutions may be uneven, which can cause the bidirectional search to fail to find effective solutions. Despite this, LSMOF outperforms WOF in some problems, indicating that finding effective search directions in the decision space is a new direction for problem solving. In this framework, PESA-II [44] employs a grid density estimation method to select Pareto solutions, successfully transforming the problem in the multi-objective space into a single-objective optimization problem. MSOPS [45] adopts a selective optimization strategy to convert multi-objective problems into single-objective problems, enhancing the selection efficiency of solutions. RVEA [46] utilizes reference vectors to guide the search, simplifying multi-objective problems into the optimization process of reference points. Meanwhile, MOEA/D-DA [47] integrates adaptive mechanisms to further simplify problem handling by dynamically adjusting optimization parameters. These methods collectively provide new insights for improving the solution efficiency of multi-objective optimization problems. The third strategy utilizes decision variable analysis (DVA) to identify useful characteristics of decision variables in large-scale multi-objective problems (LSMOPs). In MOEA/DVA [11], decision variables' convergence and divergence properties are sampled through random perturbation, dividing the decision variables into convergence variables (CV), distance variables (DV), and mixed variables (MV). Based on the definition of inseparable functions, the dependency relationships between variables in CV are identified, further subdividing CV into independent convergence variable subsets. Finally, the MOEA/D framework optimizes the problem in groups.