U-NSGA-III in Figure 3 a) and c) shows a more gradual change in colour and did not reach the maximum values for higher dimensions, indicating a slower rate of convergence and poorer HV improvement, respectively, which scale with dimensions. In contrast, results presented in Figure 3 b) and d) for ZDT1 and ZDT2, respectively, indicate that qNEHVI converges fast at a high HV improvement, as illustrated by the bright yellow coloration which appears early and maintains this up to dim=12 with little loss in initial performance. qNEHVI, while showing superiority in overall HV score for the ZDT3 and MW7 problem, had a lower rate of convergence and maximum HV improvement as dimensions increase, illustrated in Figure 3 f) and h) by the colour gradient. Although we note that in other literature, GP models tend to perform poorly at high dimensionalities, \cite{Moriconi_2020,eriksson2021high} this was not observed here, to the limit of 12 dimensions. We believe that the underlying stochastic QMC sampling used is what drives the optimisation and hence the performance remains robust.
It should be noted that in Figure 3 e), U-NSGA-III’s HV score on the ZDT3 problem scales inconsistently with dimensionality: dim=5 shows better HV improvement (brighter colour) compared to dim=2 to 4. We attribute this to the disconnected PF being strongly affected by differences in initilisation, where entire regions can be lost as the evolutionary process fails to extrapolate and explore sufficiently. Lastly, we observe in Figure 3 g) for MW7 that U-NSGA-III performs significantly worst as compared to qNEHVI, regardless of dimensionality. The presence of more complex constraints in the problem means that many solutions are likely to be infeasible and require more iterations to evolve to feasibility according to the evolution mechanism. Infeasible solutions do not contribute to HV improvement at all, and we note that this is one of the limitations of plotting using HV as a metric, where feasibility management is not clearly reflected.