## 2 The Geometric Dilution of Precision and Genetic Algorithm
### Genetic Algorithm  
The genetic algorithm is a parameter optimization algorithm commonly used to solve complex nonlinear problems. It only requires the information of the objective function and a certain search space, and it is not sensitive to the initial value, nor constrained by the smoothness of the search space. The encoded population is the basis of evolution in the genetic operation, and the fitness function is the evaluation criterion of the algorithm. The encoded space replaces the parameter space of the problem in the iterative process of the genetic algorithm to create individuals. Finally, the globally optimal solution can be obtained by providing the calculation criteria of the objective function and the upper and lower bounds of the parameters for the target optimization problem.  
1. Encoding and generating the initial population Depending on the nature of the problem, we select the appropriate coding method and generate an initial population of \(N\) chromosomes with a defined length randomly: \[\text{pop}_{i}(t),\quad t=1,i=1,2,3,\cdots,N\] (19)
2. Calculate the fitness value for each chromosome Calculate the fitness of each chromosome \(\text{pop}_{i}(t)\) in the population \(\text{pop}(t)\): \[f_{i}=\text{fitness}(\text{pop}_{i}(t))\] (20)
3. Determine whether the algorithm has converged. If so, output the search results; otherwise, continue with the following steps;
4. Perform selection The fitness value of each individual determines the likelihood of selection: \[P_{i}=\frac{f_{i}}{\sum_{i=1}^{N}f_{i}},\quad i=1,2,3,\cdots,N\] (21)  
Using the probability distribution given by Equation (21), we randomly sample some chromosomes from the current generation population \(\text{pop}_{i}(t)\) and pass them on to the next generation population, forming a novel population.  
\[\text{newpop }(t+1)=\Big{\{}\text{pop }_{j}(t)|j=1,2\cdots N\Big{\}} \tag{22}\]
5. Perform crossover We mate the chromosomes with probability \(P_{c}\) and obtain a population of N chromosomes \(\text{crosspop}(t+1)\).
6. Perform mutation With a small probability \(P_{m}\), we mutate the genes of a chromosome to form a new population \(\text{mutpop }(t+1)\). This new population is the offspring of the completed genetic operation, denoted \(\text{pop}(t)=\text{mutpop}(t+1)\), which in turn serves as the parent of the next genetic operation, and we return it as 2.  
We can use different methods to terminate the algorithm. The simplest termination condition is to specify the maximum number of genetic (iterative) generations, and stop the algorithm when it reaches this limit. We output the best individual in the current population as the optimal solution of the problem. Another judgmental method is to terminate the algorithm by using a certain criterion to determine that the population has reached maturity and has no further evolutionary trend, such as when the variance of the adaptations of all of the individuals in the population, when the difference between the average adaptations of consecutive generations is less than a very small threshold, or when the optimal value meets the desired accuracy. A similar method is to terminate the algorithm when either the variance of the fitness of all individuals in the population or the difference between consecutive generations is less than a very small threshold, or when the best value satisfies accuracy. Table 1 shows operating parameters of the basic genetic algorithm.