# Distributed Localization of Non-Cooperative Targets in Non-Coplanar Rendezvous Processes
## 4 Numerical Simulation and Analysis  
### Evaluation of Improved NSGA-III Algorithm  
To evaluate the performance of the improved NSGA-III algorithm against the traditional NSGA-III algorithm, consistent algorithm parameters were set, and the standard multi-objective test function DTLZ1 [25] was utilized for testing, as shown in Equation (18), where \(M\) is the number of objectives, \(n\) is the number of decision variables, and \(k\) is set to 5. To ensure that the population was uniformly distributed across reference points, a population size of 56 was chosen with a maximum of 500 iterations. The traditional algorithm applied a mutation distribution index of 15, while the improved algorithm had maximum and minimum mutation distribution indices of 25 and 5, respectively. \[\left\{ \begin{array}{l}f_{1}(x)=\frac{1}{2}x_{1}x_{2}(1+g(x))\\ f_{2}(x)=\frac{1}{2}x_{1}(1-x_{2})(1+g(x))\\ f_{3}(x)=\frac{1}{2}(1-x_{1})(1+g(x))\\ g(x)=100[(n-M+1)+\sum\limits_{i=M}^{n}{((x_{i}-0.5)^{2}-\cos(20\pi(x_{i}-0.5))) }]\\ x_{i}\in[0,1],M=3,n=M+k-1,k=5\end{array} \right. \tag{18}\]  
According to [25], the real Pareto front of the three-dimensional DTLZ1 problem is a plane in three-dimensional space, where \(x+y+z=0.5\). The 56 reference points are evenly distributed on this plane, as shown by the green symbols in Figure 6.  
Figure 6 illustrates the distribution of the solutions obtained from both algorithms compared to the true Pareto front of DTLZ1. The traditional algorithm, due to insufficient search capability, did not converge to the true Pareto front before the end of its iterations. In contrast, the improved NSGA-III algorithm demonstrated significantly superior performance, with its solution set closely overlapping the true Pareto front. Evaluating the results using the criteria from reference [26], the final convergence score of the improved NSGA-III was calculated as 0.0027, representing the average Euclidean distance between the results and the true Pareto front. To illustrate the superiority of the NDX operator, two parent individuals (0.3 and 0.7) were selected, and both NDX and the traditional SBX operator were applied to generate 1000 offspring within the [0,1] interval. The offspring distributions were depicted in Figure 7, revealing that offspring generated with the SBX operator clustered around the intervals [0.2-0.4] and [0.6-0.8], which are close to the parent individuals, while those produced with the NDX operator were more evenly distributed across the entire range. This indicates that the NDX operator possesses a broader search space and significantly outperforms the traditional SBX operator. Figure 6: Comparation of two algorithms’ simulation results for DTLZ1. (a) Traditional NSGA-III. (b) Improved NSGA-III. To demonstrate the effect of the cosine-like mutation distribution index NSGA-III, a proximity parameter calculated by Equation (19) was defined. The DTLZ1 function was again applied to compare the convergence speeds of the two algorithms. The proximity parameter reflects the closeness of each iteration's result to the true optimal solution; smaller values indicate greater proximity, suggesting higher optimization efficiency and accuracy. As shown in Figure 8, both algorithms reached global optimal positions by the end of the iterations. However, with the variable mutation distribution index, the improved algorithm converged by the 260 th generation, while the traditional algorithm required 360 generations to converge, demonstrating a 100-generation acceleration in convergence speed without sacrificing accuracy. \[d_{i}=\sqrt{\sum_{j=1}^{M}\left(f_{i}(j)-f_{best}(j)\right)^{2}} \tag{19}\]  
where \(d_{i}\) is the parameter of the degree of proximity of the \(i\)-th iteration, \(f_{i}\) is the optimal solution of the \(i\)-th iteration, \(f_{best}\) is the global optimal solution of the whole iteration, and \(M\) is the dimension of the objective function.