# Multi-Objective PSO with Variable Number of Dimensions for Space Robot Path Optimization
## 4 Results and Discussion
### Parameter \(p_{\text{g}}\) Influence  
The only new control parameters of VNDMOPSO compared with the initial algorithm MOPSO [36] are the two probabilities of dimensional transition to the dimension of global best, \(p_{\text{g}}\), and personal best, \(p_{\text{b}}\). Their values determine the rate of the change of particle's dimension. Thus, they can balance the exploration vs exploitation dilemma that arises whenever any EOA or HA is used. By the proper setting of \(p_{\text{g}}\) and \(p_{\text{b}}\), we can either prioritize the speed of convergence or, on the contrary, emphasize the certainty of finding a correct solution. By setting values \(p_{\text{g}}\) and \(p_{\text{b}}\) close to zero, the particles can change their dimension only sporadically, which maintains the dimensional diversity of the swarm. On the other hand, the particles start to form clusters of a certain dimension very soon and thus search it much more carefully, with the risk that the global optimum resides in the abandoned dimensions. This behavior can be well documented by the results shown in Figure 6. Here, the sizes of the clusters containing agents with a certain value of TPs, \(N_{\text{T}}\), are shown against the iterations for a problem, \(\mathcal{P}4\), having a discrete Pareto front  
Figure 5: Results of dHV metric for algorithms VNDMOPSO, VLMOPSO, and VNDGDE3 for dimensionality settings \(\mathcal{S}3\): \(N_{\text{d}}\in\{2,3,\ldots,52\}\).  
consisting of solutions from three different dimensions. While in the first case in Figure 6a, the parameters \(p_{\rm g}=0.05\) and \(p_{\rm b}=0.10\) are set robustly, the individual particles change their dimension very gradually, which allows a careful search of the individual dimensions and results in a Pareto front with three solutions having three different dimensions. In the second case, shown in Figure 6b, with \(p_{\rm g}=0.20\) and \(p_{\rm b}=0.40\) set rather to speed up the convergence of the VNDMOPSO algorithm, the algorithm finds a solution in the form of a two-element Pareto front, where the solution with dimension \(N_{\rm T}=3\) is missing. This is caused by the fact that a correct combination of a six-element DSV is the most difficult one to find.  
Based on our experience with the algorithm, it is a good strategy to set \(p_{\rm b}=2p_{\rm g}\), as recommended in Table 1. In that case, the dimensional diversity is enhanced, since the personal best dimension, \(N_{\rm b}\), and the particle's dimension, \(N_{p}\), are the same. Moreover, only one parameter (\(p_{\rm g}\)) needs to be set by the user.  
The influence of the \(p_{\rm g}\) on the quality of the solution found by VNDMOPSO is shown in Figure 7. There, the heat plots of the dHV metric are shown for all the benchmark problems and \(p_{\rm g}\) values ranging from interval \(\left\langle 0.1;1.0\right\rangle\). Please note that the dHV metric values are normalized according to obtained values for every individual benchmark problem so that they can be visualized in a single figure. From the results in Figure 7, it can be seen that, for more complex tasks such as those from the DTLZ family (taking into account the results in Figures 3-5), VNDMOPSO converges better for \(p_{\rm g}>0.8\). The algorithm converges best for less complex tasks such as ZDT1, ZDT4, and LZ1 for \(p_{\rm g}<0.2\). Concerning the intended application (robot pathfinding, where we prefer a higher certainty of finding the correct solution over the convergence speed), we recommend a compromise value of \(p_{\rm g}\leq 0.30\).  
Figure 6: Percentage of agents with different dimensions expressed in terms of the number of turning points, \(N_{\rm T}\), against the number of iterations on \(\mathcal{P}4\) problem instance for two settings of parameters, \(p_{\rm g}\) and \(p_{\rm b}\): (**a**) \(p_{\rm g}=0.05\) and \(p_{\rm b}=0.10\); and (**b**) \(p_{\rm g}=0.20\) and \(p_{\rm b}=0.40\).