# Constrained Multi-Objective Optimization of Simulated Tree Pruning with Heterogeneous Criteria
## 2 Materials and Methods
### Optimization Method  
In this paper, pruning optimization is treated as a discrete combinatorial optimization problem, where the task is to find the set \(\mathcal{X}\) of non-dominated pruning solution vectors with respect to a set of objectives \(f=(f_{1},f_{2},f_{3})\). Objective \(f_{4}\), defined in Section 2.3, is itself not an interesting target of optimization, since it can be maximized trivially by no pruning at all. Therefore, we propose a special role for objective \(f_{4}\) in order to constrain the search to certain regions of the solution space. For instance, we may prescribe that no more than 10% of the tree's biomass should be removed, which could be imposed by the constraint \(f_{4}>0.9\). This is especially important in order to regulate aggressive pruning stimulated by objective \(f_{1}\). Adherence to constraint objective bounds is implemented in a soft manner by marking the violating solutions as dominated. Such approach does not completely reject borderline solutions, making it possible to use them for the derivation of better successors. We use only \(f_{4}\) as a constraint objective in this study, but other metrics could be used. The distinctive property of a constraint objective is that it evaluates the pruning solution globally, and not with respect to individual cuts. In general, both the lower and the upper bounds for a constraint objective can be specified by the user. The use of constraint objectives to specify feasible regions in an auxiliary objective space is a new addition to the multi-objective pruning optimization framework, which complements heuristic constraints implemented in a decision space. The latter have been introduced by [PERSON] et al. (2013) in order to reduce the size of the solution space and make the search more efficient. This is achieved by two mechanisms:  
* By limiting the cut locations to those with certain local properties. In our tests, the following restrictions were enforced:
* Cutting was limited to the wood at most \(A_{\max}\) years old, in order to prevent pruning of scaffold limbs. * Pruning was restricted to locations following a branch fork that result in removal of at least \(m_{\min}\) metamers. * By constraining the dimension of the solution vectors (i.e., the number of cuts) to a range \([d_{\min},d_{\max}]\). The high-level concept of the proposed multi-objective pruning optimization is outlined in Figure 4 and Algorithm 1. It is an adaptation of a general evolutionary process  
Figure 3: Convex hull \(\mathcal{H}\), its bounding cylinder \(\mathcal{C}\), and the center of gravity \(c\) (yellow sphere) for a tree. for the construction of Pareto front approximation. Given a tree model \(\mathcal{T}\) as input, a list \(\mathcal{I}_{\text{cut}}(\mathcal{T})\subset\mathcal{I}(\mathcal{T})\) of potential cut locations is first built according to the heuristic constraints described above. From this list, the initial population \(\mathcal{P}^{(0)}=\{\mathbf{x}_{i}^{(0)}\}\) of \(\mathit{Ps}\) solution vectors is generated. Each solution vector is obtained by sampling \(d\)-times randomly and without replacement from \(\mathcal{I}_{\text{cut}}(\mathcal{T})\), where \(d\in[d_{\text{min}},d_{\text{max}}]\). The population size \(\mathit{Ps}\) is a method meta-parameter.