# Constrained Multi-Objective Optimization of Simulated Tree Pruning with Heterogeneous Criteria
## 4 Discussion  
The experiments demonstrate that diverse sets of non-dominated solutions can be obtained by using heterogeneous pruning objectives, which is valuable for both educational and analytical purposes. However, algorithmic pruning optimization has wider potential applicability to complement rule-based solutions in the developing field of automated pruning [37]. The progress of scanning technology and computer vision allows producing  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Run**} & \multirow{2}{*}{**Method**} & \multicolumn{6}{c}{**Tree Model**} \\ \cline{3-8}  & & **Figure \(\mathbf{\mathbf{\mathbf{\mathbf{  
increasingly faithful digital reconstructions of real trees [38; 39; 40]. A wider availability of relatively low-cost devices equipped with LiDAR scanners should enable further advance of the field, as already demonstrated by some recent research [11; 41]. In such scenarios, the multi-objective assessment of pruning effects may need to rely on estimation of tree parameters that are not fully observable, or cannot be captured to a sufficient level of accuracy. The main limitation of the proposed methodology is the assumption of exact knowledge of tree properties, which are provided by the simulation environment. In order to account for the possibility of missing information, fuzzy objective measures should be incorporated into the optimization process. This would also improve the alignment of the methodology with the human perception of achieved pruning goals. Another practical drawback of current implementation is its run-time, which needs to be improved for the use of methodology in interactive sessions. The differences between the optimization methods in this respect are small, because the run-time is governed by objective evaluations. The definitions of objectives in Section 2.3 indicate that their evaluation should increase approximately linearly with problem size, i.e., the number of tree internodes. The objective \(f_{1}\) in Equation (4) sums over the flower buds, whose number is linear in the number of internodes. The most time consuming part of the objective \(f_{2}\) computation is the generation of a convex hull. This can be done in \(O(nh)\), where \(n\) is the number of branch tips and \(h\) is the number of points in the hull. While \(n\) increases approximately linearly with the tree size, \(h\) was determined to be within a small constant factor for all trees. Objectives \(f_{3}\) and \(f_{4}\) are directly related to the number of internodes, so their computation is also linear. The linear relationship of run-time and problem size has been experimentally confirmed, as presented in Table 5. The small run-time differences between methods arise from their average time complexities, which are \(O(N^{2})\) for NSGA-II, \(O(N^{2}\log(N))\) for SPEA2 and \(O(N^{2}+NM)\) for MOEA/D-EAM, where \(N\) is the population size and \(M\) is the archive size. Further optimization and parallelization of objective value computation is one of priorities for future framework development. The presented use of additional objective constraints to prevent overfitting of pruning on actual target objectives is a generally applicable concept. It can be used in other MO problems where the bounds are hard to specify on optimization objectives directly. In the proposed approach to pruning optimization, the objective constraints complement heuristic decision constraints in the search space. However, the violations of the latter can be easily detected and corrected within the optimization loop. In this way the waste of computational resources on evaluation of infeasible solutions can be avoided. Violations of objective constraints, on the other hand, can only be established after their evaluation, and it is usually not clear how to resolve them. A strategy for handling the infeasible solutions in the population is therefore required.