# Multi-Objective PSO with Variable Number of Dimensions for Space Robot Path Optimization
## 4 Results and Discussion
### Benchmark Problems  
The results of the VNDMOPSO are first validated using the available benchmark problems with known Pareto fronts and corresponding Pareto-optimal sets. There is a large set of benchmark metrics to assess the quality of the found solution of a multi-objective benchmark problem. Some of them are focused on how close the found non-dominated solutions (members of \(\mathcal{E}\)) are to the true Pareto front members, \(\mathcal{PF}\), in the objective space, e.g., the generational distance [39]. Other metrics, e.g., the spread [7], express the uniformness of the found non-dominated set over the true \(\mathcal{PF}\). The commonly accepted way to express both of these requirements for a multi-objective algorithm using a single numerical value is a metric called hypervolume (HV), introduced in [40]. It determines the portion of the objective space volume that is dominated by the found set of non-dominated solutions, \(\mathcal{E}\). The hypervolume is computed as the union of volumes between the individual members of \(\mathcal{E}\) and the reference point [41]:  
\[\mathrm{HV}^{\mathcal{E}}=\cup_{i=1}^{|\mathcal{E}||}V(\mathbf{e}_{i}). \tag{8}\]  
Here, \(V(\mathbf{e}_{i})\) denotes the volume dominated by the \(i\)-th member of the non-dominated set, \(\mathcal{E}\), i.e., the volume of a hyper-cube with opposite points, \(\mathbf{e}_{i}\), and the reference point, \(\mathbf{p}\). The so-called nadir point (the collection of the worst values of individual objectives from individual extreme points of the true \(\mathcal{PF}\)[7]) is usually selected as the reference point. The larger the HV is, the better the result of the MO algorithm has been found. The only disadvantage of HV is that it's value depends on the selection of the reference point. Therefore, the distance hypervolume metric was introduced to overcome this problem:  
\[\mathrm{dHV}=\mathrm{HV}^{\mathcal{PF}}-\mathrm{HV}^{\mathcal{E}}. \tag{9}\]  
It is computed as the difference between the hypervolume dominated by the true Pareto front, \(\mathrm{HV}^{\mathcal{PF}}\), and by the found one, \(\mathrm{HV}^{\mathcal{E}}\). Thanks to this method of calculation, the influence of the choice of the reference point is neglected. The suite of benchmark problems used in this study contains 21 VND multi-objective problems whose definition was introduced in [32; 42]. These benchmark problems are based on families of well-known test suites such as DTLZ [43], ZDT [44], LI [42], and LZ [45]. The modified benchmark problems have a feature that different parts of the Pareto front can be occupied by DSVs with different dimensions. The full definition of the benchmark problems is beyond the scope of this paper. However, the method to determine the optimal dimension over the Pareto front and definitions of all the benchmark problems are summarized in the supplementary material S1. We compare the convergence properties of the novel VNDMOPSO algorithm with VLMOPSO and VNDGDE3 on the full benchmark suite using three different dimension settings to see how the convergence of the algorithms scales with the dimensionality of the problem. The dimensionality settings are summarized in Table 3. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Symbol** & **Explanation** & **Value Range** \\ \hline \(w\) & Inertia weight & Decreasing from 0.9 to 0.4 \\ \(c_{1}\) & Cognitive learning factor & 1.5 \\ \(c_{2}\) & Social learning factor & 1.5 \\ \(P_{\mathrm{g}}\) & Probability of adapting to dimension of global best & 0.02 \\ \(p_{\mathrm{b}}\) & Probability of adapting to dimension of personal best & 0.04 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Default values of VNDMOPSO controlling parameters. The results in form of boxplots of the dHV metric are shown in Figures 3-5. The results from 100 independent runs of individual algorithms were compared using the [PERSON]'s ranked sum test. Please note that the test was performed at the significance level \(\alpha=0.05\).