## 5 Boom Configuration Optimization
### Optimization Approaches
#### 5.2.2 Metaheuristic Optimization Approach  
The second approach to solve the boom configuration optimization problem utilizes an Evolution Strategy, a metaheuristic. To be able to apply these kinds of methods to the boom configuration optimization, two issues have to be dealt with:  
1. Convert the constrained optimization problem into an unconstrained one. 2. Overcome run time issues due to expensive evaluations of the structural constraints by using surrogate-assisted optimization. Unconstrained Optimization Problem  
Since metaheuristic algorithms often do not handle constraints explicitly, they have to either be incorporated in the objective function, e.g., by using _penalty functions_[45], or be addressed in the encoding of the variables, e.g., by allowing only valid solutions regarding those constraints, yielding an unconstrained optimization problem. For the BTB optimization problem, a penalty for the constraints in Equations (5)-(7) is designed and added to the objective function. The new objective function is  
\[\tilde{f}(\mathbf{x},\mathbf{y},\mathbf{z}) = f(\mathbf{x},\mathbf{y},\mathbf{z}) \tag{16}\] \[+ r\sum_{i=1}^{n}\left(\sum_{j\in\{\mathsf{b},\mathsf{s},\mathsf{ t}\}}\max(0,\sigma_{i,j}-1)+\sum_{j\in\{\mathsf{b},\mathsf{s},\mathsf{t}\}}\max(0, \phi_{i,j}-1)+\sum_{k\in\{\mathsf{b},\mathsf{s}\}}\max(0,\beta_{i,k}-1)\right)\]  
with \(r\) as constant penalty factor which has to be set in the optimization algorithm. By defining the penalty functions as the maximum of zero and the constraint, e.g., \(\sigma_{i,j}-1\), it is ensured that only for invalid solutions a penalty is added. The other constraints are taken care of in the solution encoding. Towards this end, we use an integer-vector of length \(5n+2\), containing all the variables of the boom configuration. By only selecting values from the given sets of thicknesses as well as number and type of stiffeners, it is ensured that the constraints in Equations (8)-(10), (12), (13) and (15) are fulfilled. To ensure that the constraints in Equations (11) and (14) are not violated, the thicknesses and the number of stiffeners are sorted in descending order from the pivot- to the head-end. We opt for an Evolution Strategy (ES), mainly because ES is traditionally specialized on fix-length vectors. Because we use an integer-vector, the values of the vector after manipulation are rounded to the next feasible integer. Run-Time Reduction By Surrogate-Assisted Optimization  
One common technique to reduce the computation time is _surrogate modeling_[46; 47] or, in the case of optimization, _surrogate-assisted optimization_[48]. The main idea is to replace the original, expensive evaluation \(\tilde{f}(\mathbf{x},\mathbf{y},\mathbf{z})\) (Equation (16)) with a simple model \(\hat{f}(\mathbf{x},\mathbf{y},\mathbf{z})\). To create surrogate models, machine learning techniques [49] are used. These methods use data of previously performed expensive evaluations as input to learn a model that estimates the objective value of unseen evaluations. The obtained models are not as precise as the simulation model but can be evaluated fast and are accurate enough to guide the optimization algorithm to promising designs. However, the final results must be verified with the expensive evaluation in case the optimization was misled with an improper surrogate model. In general, there are two ways of applying surrogate models with optimization. First, in offline or sequential modeling [50], a single surrogate model is obtained before the actual optimization and not changed during the optimization. Second, in online or adaptive modeling, modeling is also performed during the optimization to adapt the model during the search process. In this work, we used an offline model for optimizing the boom configuration. To train the surrogate model, we sampled the search space and evaluated the samples with the structural analysis tool to obtain reference data.