# Constrained Multi-Objective Optimization of Simulated Tree Pruning with Heterogeneous Criteria
## 3 Results
It can be observed that all of the reference solutions are dominated by the majority of non-dominated individuals found by MO methods, because their positions in objective space projections are within the dominated regions behind the non-dominated fronts. As expected, the non-selective pruning to cylindrical form improves the value of objectives \(f_{2}\) and \(f_{3}\) with respect to no pruning, while the rule-based pruning increases the value of objective \(f_{1}\). However, the results of pruning optimization demonstrate that simultaneous improvement on all objectives is possible. In relative terms, the hardest improvements to make are with respect to objective \(f_{1}\). This can be explained by analyzing the behavior of that objective, which computes the average light exposure of flower buds. By pruning away branches, the number of buds that contribute to the objective value is reduced, but at the same time the light conditions for the remaining ones are improved and the denominator in Equation (4) is lowered. This  
Figure 8: Non-dominated sets obtained by MOEA/D-EAM (blue) and SPEA2 (orange) for the corresponding trees from Figure 6. The plots include reference objective vectors for no pruning (red), cylindrical pruning (green), and rule-based pruning (black). Figure 7: Non-dominated sets obtained by NSGA-II (blue) and SPEA2 (orange) for the corresponding trees from Figure 6. The plots include reference objective vectors for no pruning (red), cylindrical pruning (green), and rule-based pruning (black). gain vs. loss ratio of light intake can be highly discontinuous even for small changes in the solution vector, which results in multiple local optima that make optimization difficult. Figure 10 shows realizations of pruning vectors from different regions of the non-dominated set on the target tree model. It is informative to compare the pruning solutions that focus on maximization of individual objectives. For example, the pruning that maximizes the light objective tends to remove branches that cast or receive a lot of shadow, whereas the balance-oriented pruning promotes one-sided removal of wood in order to restore tree equilibrium. The pruning solution focusing on shape objective \(f_{2}\) advocates branch removal from multiple sides, but contains elements of the other two prunings in order to improve on objectives \(f_{1}\) and \(f3\) as well. The next question addressed with the experiments was the evaluation of constraint objective \(f_{4}\) importance for regulating the optimization. To this end, we executed the SPEA2 optimization using a relaxed lower bound 0.8 for \(f_{4}\). The comparison of non-dominated sets for the best SPEA2 run with tight and relaxed bound is shown in Figure 11a. It is evident that further numerical improvement of objective values can be achieved by relaxing the constraint. However, the solutions tend to start overfitting on one or more pruning objectives through extensive biomass removal (Figure 11b). A proper empirical selection of bounds is, thus, necessary to confine the solutions to viable regions. Figure 10: Pruning realizations for solutions from different regions of objective space. Distinguishing pruning patterns can be observed for solutions that promote light intake **(top left)**, crown shape **(top right)**, or tree balance **(bottom right)**. Figure 9: Two-dimensional projections of non-dominated sets for the tree from Figure 6a. Figure 11a shows that the effect of the constraint objective is in bounding the feasible region of the objective space. By relaxing the constraint, we allow a wider spread of individual objective values, which increases the possible distance of non-dominated fronts from the origin. Such indirect specification of feasible regions is in contrast to defining constraints in terms of optimization objectives themselves, which is more common in MO [33]. The final goal of experiments is comparison of method performance. Figure 7 suggests that SPEA2 outperformed NSGA-II consistently in our tests, and also performed better than MOEA/D-EAM in general.