# A Two-Stage Multi-Objective Optimization Algorithm for Solving Large-Scale Optimization Problems
## 2 Background
### Large-Scale Multi-Objective Optimization  
In practical applications, there are a large number of multi-objective optimization problems containing thousands of decision variables, but existing multi-objective evolutionary algorithms cannot perfectly solve these problems. Experience has shown that as the number of variables in the search space increases, the effectiveness of most MOEAs sharply decreases. In LSMOEAs based on decision variable grouping, the result of decision variable grouping will directly affect the final solution quality. Although the existing algorithms can achieve satisfactory results in most cases, there will be a lack of diversity of solutions for some problems. As shown in Figure 1, the black curve represents the ideal Pareto frontier for the bi-objective optimization problem BT6, and the gray point represents the feasible solution obtained by the algorithm in the current optimization stage. It can be seen from Figure 1 that as the algorithm evolves, the solution obtained gradually concentrates on a local point on the Pareto front. This shows that with the evolution of the algorithm, the diversity of the population has been significantly lost. [PERSON] [15] first applied the coevolution (CC) framework to evolutionary algorithms for solving large-scale multi-objective optimization problems. Recently, [PERSON] et al. proposed a multi-objective evolutionary algorithm for solving large-scale multi-objective problems, called MOEA/DVA [22]. The algorithm uses a method called decision variable analysis to divide all decision variables into three categories. By perturbing the value of some variables, if the generated solution is not dominant after the variable perturbation and is not dominated by other solutions, the variable is called a distance variable. If there is a dominant relationship between the solutions generated after the disturbance, this variable is called the position variable. Otherwise, this variable is called a mixed variable. [PERSON] [16] proposed another new evolutionary algorithm using the decision variable clustering method. It first clusters all decision variables into two categories, one of which affects the convergence performance, and the other variables are related to diversity. Then in the optimization process, two different strategies are adopted for the two categories. In addition, there is a dynamic grouping strategy. In the paper [23], [PERSON] first designed a decomposition pool with different group sizes. Then, select one of the group sizes from the pool. Divides the entire dimension into many groups of different sizes based on the selected grouping size. This method can save a lot of computing resources. In addition, another strategy is to transform the original LSMOP into a simpler MOP with a relatively small number of decision variables. The weighted optimization framework (WOF) [24] is representative of this category. In WOF, decision variables are divided into several groups, each of which has a weight variable. Therefore, the optimization of the same group of weight variables can be regarded as the optimization of the subproblem in the original decision space subspace. Another typical algorithm is a large-scale multi-objective algorithm based on problem reconstruction (LSMOF) [25]. The core idea of LSMOF is to transform the original LSMOP into a low-dimensional single-objective optimization problem through problem reconstruction. In the reconstructed problem, the decision space is reduced from the dimension of weight variable reconstruction, and the target space is therefore reduced, so a group of excellent solutions can be obtained efficiently. There are also some other strategies, such as the recently proposed multi-objective particle swarm optimization algorithm based on the competition mechanism (CMOPSO) [26], which does not use a grouping strategy or decision variable analysis strategy but uses a particle swarm optimizer (PSO) to enhance the diversity of the population and uses a paired competition strategy [27] to solve LSMOP. [PERSON] et al.