## 3 Differential Evolution Integration Algorithm Based on Mixed Penalty Function Screening Criteria
### The Processing of the Constraint Conditions  
In the constraint optimization problems shown by \((1)\), there are two kinds of constraints: equality constraints and inequality constraints. Inequality constraints tend to expand the searching range, which is beneficial to global searching and can reduce the probability of fault tolerance. However, The equality constraints present a one-dimensional linear region, which can narrow the optimal range and is not conducive to global optimization. For this reason, we conduct high-dimension processing on the equality constraints:  
\[g_{i}=|h_{i}|-\varepsilon_{i}\leq 0, \tag{14}\]  
where \(\varepsilon_{i}\geq 0\) is a sufficiently small real-valued slack variable, which increases the searching range by expanding the dimension, which is beneficial to global searching and global convergence. However, the phenomenon will reduce the feasible domain searching ratio, which is not beneficial to finding the global convergent point \(x^{*}\). To solve the problem, we adopt the method of the self-adaptive slack variable to deal with the equality constraints [43]. \[If(R_{n}(k)\leq R_{l}),\textit{Then }\varepsilon_{i}(t+1)=\beta_{l} \varepsilon_{i}(k) \tag{15}\] \[If(R_{n}(k)\geq R_{u}),\textit{Then }\varepsilon_{i}(t+1)=\beta_{u} \varepsilon_{i}(k), \tag{16}\]  
where \(R_{n}\) is the proportion of feasible solutions satisfying the slack variable in the current population, \(0\leq R_{l}\leq R_{u}\leq 1,0\leq\beta_{l}\leq 1\leq\beta_{u,}\varepsilon_{i}(0)\) is the maximum value that violates the effective solution in the initial population. Implementation of Differential Evolution Integrated Algorithm Based on the Screening Criterion of the Mixed Penalty Function  
In this paper, the mixed penalty function screening criterion is applied to the DE algorithm, the validity and the convergence of the algorithm are analyzed by the Markov model. Finally, we obtain the differential evolution integrated algorithm based on mixed function screening criteria (DE-MPFSC algorithm). In the DE-MPFSC algorithm, let the space dimension be \(D=n\), the population individuals are a set of \(S_{X}=\{(x_{i}^{t},\delta_{i})|i=1,2,\cdots,NP\}\), \(x_{i}^{t}\) is an \(n\)-dim decision variable, \(\delta_{i}\) is the \(n\)-dim individual iterative step variable. The initial population is evenly distributed in the \(n\)-dim space constraints. The mutation operator, crossover operator and selection operator \(DE\) algorithm can act on the \(n\)-dimspace constraints and work on all individuals in the initial population to improve their searching ability, where the searching step calculation formula of the DE-MPFSC algorithm is as follows:  
\[\delta_{i}^{t+1}=\delta_{i}^{t}+exp(\tau\cdot N(0,1)+\tau^{\prime} \cdot N_{i}(0,1)) \tag{17}\] \[x_{i}^{t+1}=x_{i}^{t}+\delta_{i}^{t+1}\cdot N_{i}(0,1), \tag{18}\]  
where \(t\) is the number of iterations, and \(\tau\) and \(\tau^{\prime}\) are the self-adaptive learning rate of the population individuals. To ensure the accuracy of the calculation, we calculate according to the method proposed by [PERSON] [44]: \(\tau=(\sqrt{2\sqrt{n}})^{-1}\) and \(\tau^{\prime}=(\sqrt{2n})^{-1}\). \(N(0,1)\) and \(N_{i}(0,1)\) are all real uniform Gaussian distributions with a mean of 0 and a variance of 1. Then, we select individuals in the algorithm according to the mixed penalty function screening criteria and keep the outstanding individuals of \(\zeta\) parent individuals and \(\zeta\) progeny individuals to the next generation. The DE-MPFSC algorithm steps are as follows:  
**STEP 1**: Initializing variables: According to the number of the population and the number of individuals, let \(t=0\), then generate the initial population within \(\psi\) parents individuals and \(\varphi\) progeny individuals and each of the individuals corresponds with \(\{(x_{i}^{t},\delta_{i})|i=1,2,\cdots,NP\}\). **STEP 2**: Calculating the fitness function value of the initial variable: Calculate the fitness function values of \(\psi\) parent individuals and record the maximum value and minimum value.