# An Evidential Reasoning Assessment Method Based on Multidimensional Fault Conclusion
## 3 An Assessment Method of Evidence Reasoning Based on Fault Conclusion
### Introduction to the CMA-ES Optimization Algorithm  
This paper utilizes an adaptive ER rule model optimized by the CMA-ES. CMA-ES demonstrates outstanding performance in handling non-convex optimization problems and high-dimensional parameter spaces [26]. It excels in exploring complex solution spaces more comprehensively, particularly when the objective function has multiple local optima, thereby facilitating the identification of global optima. Additionally, CMA-ES efficiently navigates and converges in high-dimensional spaces through adaptive adjustments of the covariance matrix, making it particularly effective for optimizing complex parameters in the ER model. The main purpose of the CMA-ES algorithm is to fine-tune the optimization parameters, especially the step size and covariance matrix, to enhance performance. These adjustments to the optimization parameters play a crucial role in the swift convergence of evolutionary algorithms. The core principle of the CMA-ES algorithm in parameter optimization is to adjust the covariance matrix, which indirectly guides the evolutionary path across the search space, thus increasing the probability of finding optimal solutions. The objective function in the adaptive ER algorithm model has two components: the monotonicity coefficient (\(\text{Mon}_{j}\)) indicator and the correlation coefficient (\(\text{Corr}_{j}\)) indicator. Initially, all equality conditions are transformed into objective functions. For solution \(\Phi\), multiple independent objective functions can be formulated, enabling the CMA-ES algorithm to address each one separately. When using the CMA-ES algorithm to address the equality constraint conditions independently, the following conditions should also be met:  
\[ \begin{split}\text{if}&\bar{\upsilon}_{j}\geq 1\\ \bar{\upsilon}_{j}=1\\ \text{if}&\bar{\upsilon}_{j}\leq 0\\ \bar{\upsilon}_{j}=0\end{split}  \tag{19}\]  
These conditions ensure that the parameters \(\bar{\upsilon}_{j}\) in solution \(\Phi\) meet the inequality constraints in the ER model. This optimization algorithm draws on the concept of multi-objective optimization algorithms, converting each constraint into a specific unconstrained objective function. Each iteration independently seeks the optimal value, ensuring the results fully comply with the established constraints, thereby efficiently solving the model optimization problem. In particular, the execution of this algorithm can be broken down into the following essential steps. Step 1: Sampling Operation  
The sampling operation focuses on selecting a solution within the solution space (comprising all ER parameter vectors \(\Phi\)) as the central point and creating an initial population based on a normal distribution. This population is denser near the central point and becomes sparser with increasing distance. In the CMA-ES algorithm, this central point represents the expectation. Here, the initial parameter vector \(\Phi^{0}\) of the ER model is chosen as the expected value. The population generation method is as follows:  
\[\Phi_{k}^{g+1}\sim mean^{g}+\eta^{g}\mathbb{N}(0,\mathbf{C}^{g}),\ k=1,2, \cdots,\lambda \tag{20}\]  
where \(\Phi_{k}^{g+1}\) denotes the \(k\)-th solution in the population, \(g\) represents the evolutionary generation, \(mean^{g}\) denotes the center expectation of the \(g\)-th generation population, \(\eta\) represents the step size for distribution update, \(\mathbb{N}(0,\mathbf{C}^{g})\) represents the multivariate normal distribution, and \(\mathbf{C}^{g}\) denotes the covariance matrix (positive definite matrix) in the \(g\)-th generation's search distribution. Geometrically, this corresponds to the elliptical distribution of the population in the solution space. Step 2: Selection Operation  
The purpose of this operation is to select solutions with higher fitness values from a pool of potential optimal candidates. Selection is determined by the fitness function value; a higher value indicates a better solution.