# Research on Multi-Objective Evolutionary Algorithms Based on Large-Scale Decision Variable Analysis
## 2 Related Work
### Decision Variable Quantitative Analysis Method Based on PBI  
To tackle the issue of missing quantitative analysis mechanisms for variable characteristics in existing decision variable grouping methods, this paper proposes a decision variable quantitative analysis method (DVQA) based on Penalty Boundary Intersection(PBI). This method utilizes the convergence and divergence trends of subpopulations to accurately represent the convergence and diversity levels of decision variables, thereby quantifying their behavior in the optimization process. This provides more accurate and comprehensive decision support for optimization algorithms, enhancing the performance and efficiency of multi-objective optimization algorithms.  
Based on the study of different control attributes of decision variables and the PBI method, the proposed DVQA method employs distinct optimization strategies for variables with varying attributes during the population evolution process. The obtained characteristic values of the variables are utilized in the directed crossover and mutation mechanisms of the population. This aims to enhance the convergence efficiency of the population, thereby further improving the overall performance of multi-objective optimization algorithms.  
Let \(X_{\text{far}}^{\lambda}\) be the point in the perturbed subpopulation that is farthest from the hyperplane \(HP\), and \(X_{\text{near}}^{\lambda}\) be the point that is closest to \(HP\). The vector \(z^{*}=z^{\text{nadir}}-z^{\text{ideal}}\) represents the normal vector in the subpopulation, while \(W\) denotes the global normal vector. By connecting the farthest point from the hyperplane to the nearest point, and then projecting this line onto the global normal vector, we obtain the convergence trend \(C_{x}\) of variable \(x\), as shown in Equation (1). This allows us to assess the proximity of the subpopulation to the ideal solution. This is because, in PBI, the normal vector represents the ideal direction in the objective space, and the computed projection distance reflects how the subpopulation clusters with respect to the optimization objectives. A smaller distance indicates that the solutions are closer to the ideal point and exhibit a higher degree of convergence. Let \(X_{1}^{S}\) and \(X_{2}^{S}\) be the two points in the population that are farthest apart on the hyperplane, with the projection distance between these points on the subpopulation hyperplane representing the diversity trend \(D_{x}\) of variable \(x\), as shown in Equation (2). Here, the projection result indicates the extent of the population's spread on the hyperplane. A longer projection length suggests that the subpopulation is widely distributed in the objective space, demonstrating good diversity. Equation (3) demonstrates the differences between the convergence and diversity metrics. When \(R_{x}<0\) or \(|R_{x}|<\theta\), the decision variable \(x_{i}\) is classified as a diversity-related variable, with a diversity degree of \(|R_{x}|\); when \(R_{x}>0\), the decision variable \(x\) is classified as a convergence-related variable, with a convergence degree of \(R_{x}\). Algorithm 2 presents the pseudocode for the proposed DVQA algorithm.  
\[C_{x}=\left\|\frac{(x_{\text{far}}^{\lambda}-x_{\text{near}}^{\lambda})\cdot \mathbf{z}^{*}\cdot\mathbf{w}}{\|\mathbf{z}^{*}\|\|\mathbf{w}\|}\right\|, \tag{1}\]  
\[D_{x}=\left\|(X_{1}^{S}-X_{2}^{S})-\frac{(X_{1}^{S}-X_{2}^{S})\cdot\mathbf{z} ^{*}}{\|\mathbf{z}^{*}\|}\right\|, \tag{2}\]  
\[R_{x}=C_{x}-D_{x}. \tag{3}\]