## 4 Optimization Approach
### Two-Level PSO-Based Approach to Aircraft Climb Path Optimization  
Criticism over population-based optimization methods mainly focuses on the excessive number of fitness function evaluations required for locating the optimal solutions: Although these methods are very capable of conducting a global search in the optimization domain, in applications where the computational cost per evaluation is considerable, the optimization turnaround time becomesexcessive. For a preset number of fitness function evaluations, in some cases this results in sub-optimal, non-converged solutions. To remedy this problem, two options are generally available:  
1. A reduction in the number of design variables. 2. The use of a surrogate model [37; 38] to better initialize the search or filter-out non-promising candidate solutions. In this article, in order to introduce a computationally competitive climb path optimization methodology, both strategies were adopted: As specified in Section 2, Bezier splines were used for the generation of climb paths, reducing design variables to the coordinates of a finite number of control points, rather than solving the original highly dimensional optimal control problem; the use of a parameterized curve is a common feature with other, similar methods, however, these are hereby used to design trajectories and not control sequences, facilitating the selection of inputs. Furthermore, E-M predictions are used as a surrogate model of the actual cost functions in a proposed two-level optimization strategy. This focuses on reducing the turnaround time of the simulation-based optimization run by pre-evaluating the problem in the E-M domain: An initial low-level optimization run is performed using E-M as a low-cost, low-fidelity approximate of the actual objective functions. \(TTC\) and \(FTC\) are obtained from numerical integration of the quantities:  
\[TTC=\int_{E_{s_{1}}}^{E_{s_{2}}}\frac{1}{P_{s}}dE_{s} \tag{32}\]  
\[FTC=\int_{E_{s_{1}}}^{E_{s_{2}}}\frac{\dot{m}_{f}}{P_{s}}dE_{s} \tag{33}\]  
along candidate flight paths, \(m_{f}\) accounting for fuel flow rate. Whereas \"traditional\" E-M considers trajectory optimization only in terms of time and fuel burn, it is evident that a simple modification of Equation (33) (in particular, a replacement of \(m_{f}\) with another appropriate measure) can be used to generate cost functions for other quantities that have recently attracted scientific interest, such as noise and pollutant emissions. For reasons of simplicity, in the present study, however, the analysis is restricted to minimum time and fuel trajectories, which are more suited to the military aircraft/engine integration application that is presented. Solutions generated at the first level are used to initialize a second, high-level optimization run which employs the aircraft simulation to accurately assess the outcome of candidate flight paths. In both levels, the MOPSO algorithm is used to conduct the search. The flowchart of the process is shown in Figure 9.  
Figure 9: Flowchart of the proposed two-level optimization scheme. The performance of the two-level optimization scheme was compared with that of standard MOPSO in a climb path optimization problem with 8 design variables (4 control points \(\times\) 2 coordinates per point) using a population of 20 particles. After some trial and error analysis, PSO constants were set at \(c_{1}=c_{2}=1.7\) and \(w=0.3\). For the two-level optimization case, an initial low-level run of 300 iterations was specified. Start conditions were set at M = 0.8, h = 0 m and end conditions at M = 1.8, h = 14,000 m. The hypervolume indicator [39] was used to compare the convergence speed of the two methods. In a two-objective problem, this equals the area of the objective space formed between the origin and a user-defined \"nadir\" point (for our study, this was set at (800, 2500)) that is dominated by the Pareto front. In order to address the randomness of the PSO, 10 optimization runs were performed for each method.