# Research on Multi-Objective Evolutionary Algorithms Based on Large-Scale Decision Variable Analysis
## 2 Related Work
### Algorithm Framework  
By studying the characteristics of decision variables, this paper analyzes the impact of convergence decision variables on each objective and uses the most affected objective as the maximum contribution objective of the decision variable. Combining the dual-population co-evolution framework, a dual-population co-evolution algorithm based on large-scale decision variable analysis (DVA-TPCEA) is proposed. The pseudocode of the algorithm is shown in Algorithm 1.  
```
0:\(D\) (Number of decision variables), \(M\) (Number of targets)
0:\(DV\) (Diversity-related decision variables), \(CV\) (Convergence-related decision variables)
1:\([P,W]\leftarrow\) Initialize(\(N\))
2:\([CV,DV,CX,DX]\leftarrow\) DecisionVariableQuantitativeAnalysis(\(D,M,P,W\))
3:\(FO\leftarrow\) ContributionTargetAnalysis(\(CV,DV,W\))
4:\(desSet\leftarrow\) GroupingBasedContributionTarget(\(CV,DV,FO,W\))
5:while termination criterion not fulfilled do
6:\([CP,DP]\gets P\)
7:for\(j=1\) to \(M\)do
8:for\(i=1\) to \(|W|\)do
9:\(SCP\leftarrow\) ConvergenceDirectionalCrossover-Variation(\(CP,W,CV,desSet\))
10:endfor
11:endfor
12:\(newCP\leftarrow\) ConvergenceEnvironmentalSelection(\(CP,SCP\))
13:for\(j=1\) to \(M\)do
14:for\(i=1\) to \(|W|\)do
15:\(SDP\leftarrow\) DiversityDirectionalCrossover-Variation(\(DP,W,DV,desSet\))
16:endfor
17:endfor
18:\(newDP\leftarrow\) DiversityEnvironmentalSelection(\(DP,SDP\))
19:\(P\leftarrow\) EnvironmentalSelection(\(newCP,newDP\)) {\(d_{1}+\theta d_{2}\)}
20:endwhile
```  
**Algorithm 1** DVA-TPCEA  
For the given large-scale high-dimensional multi-objective optimization problem, a decision variable quantitative analysis method (DVQA) based on Penalty Boundary Intersection (PBI) is designed (Algorithm 2). This method utilizes the convergence and divergence trends of subpopulations to categorize variables into convergence-related variables and diversity-related variables, deriving the characteristic values \(C_{x}\) and \(D_{x}\) for each variable. Subsequently, the contribution of convergence-related variables is calculated using the Contribution Objective Analysis algorithm (Algorithm 3) to obtain their contribution objective \(f_{m}(x)\). A grouping method is employed to cluster convergence-related variables with the same contribution objective (Algorithm 4). In the dual-population evolutionary phase, the algorithm defines separate convergence and diversity populations. During the optimization of the convergence population (Algorithm 5), a directed crossover and mutation mechanism based on the contribution grouping of convergence variables is designed, utilizing the \(d_{1}\) value (PBI) for offspring selection during the environmental selection phase. Conversely, for the optimization of the diversity population (Algorithm 6), a directed crossover and mutation mechanism based on diversity variables is implemented, with the \(d_{2}\) value (PBI) used for offspring selection in the environmental selection phase. Finally, the population undergoes environmental selection using the \(d_{1}+\theta d_{2}\) metric.