# Multi-Fidelity Gradient-Based Optimization for High-Dimensional Aeroelastic Configurations
## 3 Results
### Airfoil Design  
Figure 11 shows the single- and multi-fidelity convergence histories of the BFGS-based optimizers for the airfoil optimization cases (labeled problems P1a and P1b in Table 6) using rigid and aeroelastic analyses.  
Figure 10: Baseline and optimized geometries. For reference, the low- and high-fidelity angles of attack are 1.339148 and 0.8001988 degrees, respectively. **(a)** Yehudi; **(b)** Tip.  
Figure 9: Convergence histories of aeroelastic problem P1a using SLSQP. **(a)** Objective; **(b)** Feasibility.  
(The penalty-constrained, single-fidelity BFGS was selected as an algorithmically similar reference for MF-BFGS rather than SLSQP.) In particular, Figure 10(a) shows the penalty-constrained objective function versus the number of high-fidelity evaluations, with solid and dotted lines representing single- and multi-fidelity optimizers, respectively. This is the quantity directly minimized by the unconstrained optimizers. To illustrate how well the inequality constraints are satisfied through the penalty functions, Figure 10(b) shows the maximum constraint value for each case, which generally drops by roughly two orders of magnitude throughout the optimizations. Figure 10(c) shows the reduction in the penalty-constrained objective as a percentage of each case's initial value. As a way to quantify the performance of using multiple fidelities, Figure 10(d) shows the difference of reductions (multi-fidelity minus single-fidelity). At any given design evaluation, a positive value of this metric indicates an advantage of using multiple fidelities over the single-fidelity counterpart, since it means the high-fidelity design from the multi-fidelity optimizer is better than that of the single-fidelity for the same computational cost (assuming negligible low-fidelity model cost). Note that it will approach zero if the single- and multi-fidelity optimizers both reach the same solution.  
In general, all four cases (rigid and aeroelastic P1a and P1b) showed some benefits early in the optimization process (i.e., for around 3-10 high-fidelity evaluations). For example, at around five evaluations, three out of four cases were roughly 5-10% ahead of the single-fidelity optimizer. By 20 evaluations, the metric for some of the cases dipped below zero slightly, but generally approached zero (indicating both optimizers were approaching the same solution), with the exception of the higher-dimensional rigid case. Unfortunately, for  
Figure 11: Optimization results for airfoil design cases. In (**a–c**), solid and dotted lines represent single- and multi-fidelity results, respectively. (**a**) Penalty-constrained objective; (**b**) Feasibility; (**c**) Percent reduction, all; (**d**) Percent reduction, difference.  
this particular case, the multi-fidelity optimizer stopped at an optimum that was roughly 1.8% worse than that of the single-fidelity. It is unclear if this was due to errors in gradients, design space complexity (and constraint handling through penalty functions), or numerical noise due to the regeneration of unstructured meshes. Although this stalling could be algorithmic, it was presumed to be unlikely as the penalty-constrained SLSQP result in the previous section also appeared to reach a worse solution than SLSQP with direct constraint handling.