# Research on Multi-Objective Evolutionary Algorithms Based on Large-Scale Decision Variable Analysis
## 2 Related Work
### Research Status and Problem Analysis  
Based on the relevant research conducted by both domestic and international researchers, the current state of research and the challenges it faces can be summarized as follows:  
(1) Research Classification and Focus: In the field of multi-objective evolutionary algorithm research, LaMaOPs are typically divided into large-scale multi-objective optimization problems and high-dimensional multi-objective optimization problems for independent study. The scalability concerning the number of objectives has received extensive attention, but the scalability in terms of the number of decision variables has not been fully studied. (2) Variable Grouping Mechanisms: Most large-scale multi-objective evolutionary algorithms use variable grouping mechanisms to reduce the dimensionality of decision variables, followed by optimization using high-dimensional multi-objective evolutionary frameworks. The above research has achieved some success in solving large-scale optimization problems with 2-3 objectives, but there has been little extension to high-dimensional multi-objective optimization problems. The reasons include:  
* Information Loss: Variable grouping may cause a loss of population evolution information to some extent, enhancing convergence but reducing diversity, leading to local optima in high-dimensional MOEAs. * Lack of Quantitative Analysis: There is a lack of quantitative analysis mechanisms for the convergence and divergence characteristics of decision variables, reducing grouping accuracy and affecting search efficiency in high-dimensional decision spaces. * High Computational Complexity: In a multi-objective optimization context, traditional variable grouping mechanisms have a high computational complexity of \(O(n^{2}m)\) (where \(n\) represents the scale of decision variables and \(m\) represents the number of objective variables), resulting in poor scalability of the algorithm. (3) Balancing Convergence and Diversity: To address the difficulty in balancing convergence and diversity, researchers have optimized and mitigated this issue by modifying dominance relationships, introducing density estimation, and constructing comprehensive indicators. These methods share the common goal of maintaining population diversity and convergence during a single population optimization iteration. However, as the number of objective variables increases, the conflicts between different objectives intensify, leading to a majority of individuals in the population being non-dominated [51]. This \"simultaneous optimization in one iteration\" mechanism struggles to fundamentally resolve the conflict between convergence and diversity. In summary, while existing approaches have made strides in addressing LaMaOPs, they still face challenges in effectively analyzing decision variable characteristics, balancing convergence and diversity in large-scale settings, and addressing practical scheduling applications. To bridge these gaps, our proposed DVA-TPCEA incorporates (1) a quantitative analysis of decision variables to better utilize intrinsic variable characteristics, (2) a dual-population cooperative evolution mechanism to improve performance across diverse objectives and decision variable scales, and (3) a large-scale container resource scheduling strategy to demonstrate the algorithm's applicability to real-world problems. These contributions address the need for efficient grouping strategies and advanced quantitative analysis mechanisms, and establish an innovative framework to manage the complex interplay between convergence and diversity in high-dimensional multi-objective optimization problems. DVA-TPCEA: A Dual-Population Cooperative Evolutionary Algorithm Based on Large-Scale Decision Variable Analysis  
This section begins by presenting the overall framework of the DVA-TPCEA algorithm to illustrate the execution order. It then describes each key step in detail, accompanied by the corresponding pseudocode.