## 3 Differential Evolution Integration Algorithm Based on Mixed Penalty Function Screening Criteria
### The Screening Criteria of the Mixed Penalty Function  
When solving nonlinear constrained optimization problems \((1)\), to effectively balance the swing problem generated by the equality penalty and inequality constraints. Let the iterative point be \(x_{k}\) after \(k\) iterations, and the accuracy of the iterative solution \(x_{k}\) be a sufficiently small real value \(\varepsilon\in(10^{(-4)},10^{(-5)})\). According to the brief screening rules of the effective solutions proposed by [PERSON] (2016), we establish the screening criteria of the mixed penalty function based on the internal penalty function (this screening criterion was established to relax the spatial search area in order to facilitate cross-regional search of the objective function), the external penalty function and their normalization, as follows:  
1. When the two effective solutions \(x_{k_{1}}\) and \(x_{k_{2}}\) are inside the feasible domain \(int(D)\) under the constraint conditions, the effective solution with a smaller internal penalty function value is considered the optimal approximate solution and then \(Pr_{1}=\frac{x_{k_{1}}}{\sum_{i\in I\times k_{2}}}\), \(x_{k}\) satisfies the formula \(f(x_{k})=\min(f(x_{k_{1}}),f(x_{k_{2}}))\);
2. When the two effective solutions \(x_{k_{1}}\) and \(x_{k_{2}}\) appear in the form of certain solutions and are located at the boundary of the feasible domain \(Round(D)\) under the constraint conditions, we take the mean value of the two effective solutions as an optimal approximate solution. Then, \(Pr_{1}\frac{\min\{x_{k_{1}},x_{k_{2}}\}}{\sum_{i\in I\times i\in D}x_{k}}\), \(Pr_{2}=\frac{\max\{x_{k_{1}},x_{k_{2}}\}}{\sum_{i\in I\times i\in D}x_{k}}\);
3. When the two effective solutions \(x_{k_{1}}\) and \(x_{k_{2}}\) appearing in the form of uncertain solutions are located at the boundary of the feasible domain \(Round(D)\) under the constraint conditions, let \(x_{k_{1}}\) be an internal iterative solution, then all internal iterative solutions constitute a positive monotonic increasing sequence \(\{x_{k_{1}}\}\) within the feasible domain. In addition, let \(x_{k_{2}}\) be an external iterative solution, then all external iterative solutions constitute a positive monotonic increasing sequence outside the feasible domain. At the same time, let \(x_{k_{1}}=\inf\{x_{k_{1}}\}\), \(x_{k_{2}}=\sup\{x_{k_{2}}\}\), then we take the mean value of the two effective solutions \(\frac{\inf\{x_{k_{1}}\}+\sup\{x_{k_{2}}\}}{\sum_{i\in I\times i}x_{k}}\) as an optimal approximate solution. Further, there is the following: \(Pr_{1}=\frac{\max\{\inf\{x_{k_{1}}\},\sup\{x_{k_{2}}\}\}^{2}}{\sum_{i\in int (I)\cup i\in I}x_{k}}\), \(Pr_{2}=\frac{\max\{\inf\{x_{k_{1}}\},\sup\{x_{k_{2}}\}\}}{\sum_{i\in int(I) \cup i\
otin I}x_{k}}\);
4. When the two effective solutions \(x_{k_{1}}\) and \(x_{k_{2}}\) are outside the feasible domain under constraint conditions, the effective solution with the larger exterior penalty function value will be considered as the optimal approximate solution and then \(Pr_{2}=\frac{x_{k_{2}}}{\sum_{i\in I}x_{k}}\), \(x_{k}\) will satisfy the formula \(f(x_{k})=\max(f(x_{k_{1}}),f(x_{k_{2}}))\);
5. When the solution of the constraint is an invalid solution, in order to ensure the global nature of the algorithm search, we will perform the domain expansion operation based on the probability of the two invalid solutions: Firstly, the invalid solution \(x_{1}^{\prime}\) is taken as the center and the original space search area is dimension expanded with the preset precision \(\varepsilon\in(-1,1)\) as the radius, so that the constraint condition is relaxed and the invalid solution is validated. The screening criterion possesses the following properties:  
1. The internal penalty function or the external penalty function can only detect the inside of the feasible domain or the outside of the feasible domain; they are unable to detect the feasibility of the area by crossing the regional boundary, which can divide the feasible domain into two parts, interior and exterior.