The accuracy and error estimation algorithm of optimal solution for genetic algorithm

Absrtact: Using the genetic algorithm to solve the 0-1 programming problem, the problem of using traditional optimization method to solve the long search time is solved. However, because the kernel algorithm of genetic algorithm contains random search method, the "optimal solution" is not necessarily optimal, that is to say, in a great problem, the optimal solution of genetic algorithm may be smaller than the optimal solution obtained by using traditional enumeration algorithm. For engineering applications, the optimal solution of genetic algorithm and the approximate degree of the actual optimal solution or the estimation of the error are of great value. By analyzing the relationship between the optimal solution of general linear programming and boundary value, this paper puts forward the method of error estimation for optimal solution of genetic algorithm by "perturbation boundary Method", and discusses the reliability of perturbation boundary method. 1 Principle of perturbation boundary method

The general linear programming model can be written in the following form:

(1)

(2)

where (1) is the objective function, (2) is the constrained equation group.

According to the general linear programming principle, the optimal solution of the linear programming model must be obtained at the boundary point. In the above model, if the maximum value is obtained, it must be a boundary point, that is, the solution can make a constrained equation (without losing its generality, assuming the K-equation) satisfies the following conditions:

(3)

In general, when the K-constraint equation satisfies the above-mentioned equation, the other constraint equations generally fail to satisfy the above-mentioned equality conditions, that is, the equation J (j<>k) is:

(4)

At this time, when the constrained boundary value of the constraint equation K changes within a smaller range (), the new solution will still satisfy the (3) formula and (4) formula. In other words, when the constrained boundary value of the constraint equation K changes in (), the solution is only related to the change, that is, in (), the optimal solution is only related to the boundary value of the other constrained equations, and the equation K can be called the key equation of the optimal solution. For practical problems, the key equations of the optimal solution correspond to the scarce resources.

In the area near the boundary point of the key equation (), the relationship can be established. It can be proved that the linear relationship is:

(5)

By establishing the relationship between the optimal solution and the boundary value near the boundary point of the key equation, the method of analyzing the degree of deviation between the inexact solution and the exact solution is called "perturbation boundary method". 2 basic steps for error estimation of optimal solution

For the genetic algorithm, when the optimal solution cannot reach the boundary point, the optimal solution and the boundary value relation can be established by the method of "perturbation boundary method", and the approximation degree or error of the "optimal solution" obtained by the genetic algorithm and the actual optimal solution is estimated by the following method:

(1) Solving 0-1 problems

First, a genetic algorithm is used to solve 0-1 problems, and the optimal and corresponding constraint boundary values are obtained.

(2) Determining the constraint of the key constraint equation calculation

The relative deviation between the implementation value of each constraint boundary and the original constraint boundary is calculated, and the corresponding constraint equation with the least relative deviation is found, which is not only the key constraint equation, but the k constraint equation.

(3) Boundary value of perturbed critical constraint equation

The boundary values of the key constrained equations are small perturbation, and the boundary value after disturbance is.

(4) The optimal solution after the disturbance is calculated

The realization of the optimal solution and boundary value re-calculated by the perturbation boundary

(5) Establishing the relationship between the optimal solution and the boundary value of the key constraint equation

Use and establish linear relationships, where:

(6)

(7)

(6) Estimating the optimal solution error

Absolute Error:

(8)

Relative error:

(9) 3 Examples of optimal solution error estimates

Solve the 0-1 problem with the following constraint values:

Constraint variables	B1	B2	B3	B4	B5
Constraint value	1000	9000	23000	200	100

The optimal target solution 394.4 is obtained, and the corresponding boundary values are implemented as follows:

Scheme	B1	B2	B3	B4	B5
Constraint Implementation Value	39222.7	8888	22438	176	1083

Calculate the deviation between the implementation value of each boundary and the target boundary value, as shown in the following table:

Constraint variables	B1	B2	B3	B4	B5
Constraint implementation Value deviation	3822.3	1.2	2.4	12.0	983.0

As you can see, the b2 corresponding boundary implementation value is the least deviation from the target boundary value. The constraint equation of B2 is defined as the key constraint equation and solved.