# Research on Multi-Objective Evolutionary Algorithms Based on Large-Scale Decision Variable Analysis
## 1 Introduction  
Multi-objective optimization problems [1] are prevalent across various industrial and scientific fields, such as path planning [2], cloud resource scheduling [3], and system control [4], and can be abstracted as multi-objective optimization problems. With the development of the information age, particularly with the rise of cloud computing, big data, and artificial intelligence, the characteristics of multi-objective optimization problems have begun to shift: on the one hand, the scale of decision variables is continuously increasing; on the other hand, the number of objectives to be optimized is gradually rising [5, 6]. Under the framework of evolutionary computation, this has given rise to a new research direction: large-scale many-objective optimization problems (LaMaOPs) [7]. In this research area, researchers have proposed numerous significant works [8, 9]. Traditional multi-objective evolutionary algorithms perform excellently in solving low-dimensional problems, such as NSGA-II [10], MOEA/D [11], and IBEA [12]. When these algorithms encounter large-scale optimization problems, their performance deteriorates sharply. Multi-objective optimization problems refer to situations where two or more conflicting objective functions need to be optimized simultaneously during the optimization process. In contrast, large-scale optimization problems involve a significant numberof decision variables and constraints. When the number of decision variables exceeds 100, these optimization problems are classified as large-scale optimization problems. Typically, the number of decision variables in such problems can reach hundreds, thousands, or even more. This deterioration is primarily reflected in the exponential growth of the decision space and the intensification of optimization conflicts [13]. This is mainly due to the exponential increase in the number of possible solutions as the number of decision variables increases, making it more difficult and time-consuming for the algorithm to search the solution space [14]. As the number of objectives increases, the conflicts between them may become more severe, complicating the process of finding an optimal solution that balances multiple objectives [15]. Meanwhile, indirect encoding [16], which is a commonly used method in artificial intelligence and genetic algorithms, generates structures through indirect rules or parameters. This approach allows for a more compact representation of complex structures, reduces the search space, and enhances the scalability and generalizability of models. Consequently, indirect encoding is particularly useful in the evolution of complex structures [17]. Although various algorithms have been proposed to address LaMoDPs, such as Pareto dominance-based, decomposition-based, and indicator-based multi-objective frameworks, these algorithms often focus solely on either the scale of decision variables or the number of objectives, leading to issues such as insufficient search capability and difficulties in balancing convergence and diversity in practical applications. To provide a clear perspective on the scale addressed by the proposed method, the number of objectives and decision variables used in our experiments ranges widely to cover both many-objective and large-scale scenarios. Specifically, for the DTLZ4-7 and WFG6-9 benchmark functions, we consider cases with 3 to 10 objectives and decision variables ranging from approximately 12 to 200. For the LSMOP1-5 test functions, designed for large-scale optimization, we increase the complexity by using 3 to 15 objectives and decision variables ranging from 100 to 5000. This paper proposes a dual-population cooperative evolutionary algorithm based on large-scale decision variable analysis (DVA-TPCEA). To validate the effectiveness of the algorithm in practical problems, a large-scale container resource scheduling strategy based on DVA-TPCEA is constructed to address container scheduling issues. The main contributions of this paper are as follows:  
1.