# Multi-Fidelity Gradient-Based Optimization for High-Dimensional Aeroelastic Configurations
## 2 Modeling and Analysis
### Multi-Fidelity, Gradient-Based Design Optimization  
While global, derivative-free optimization methods such as surrogate-based or genetic algorithms are well-suited for multimodal, noisy, or non-differentiable design problems, gradient-based methods are ultimately needed when optimizing an expensive function with tens of design variables or more. For such cases, methods based on BFGS or Sequential Quadratic Programming (SQP) tend to be some of the most efficient [37], with SQP having the advantage of direct constraint handling. Such methods scale reasonably well with the number of design variables, but may still require upwards of 100 high-fidelity evaluations to reach an optimum. The use of multi-fidelity methods therefore becomes attractive if they have the potential to reduce the total number of high-fidelity evaluations required. The multi-fidelity BFGS (MF-BFGS) method presented in [13; 14] (previously termed the unified multi-fidelity or multi-fidelity quasi-Newton) is used to demonstrate gradient-based, multi-fidelity, aero-structural design optimization. The approach is an unconstrained optimization method similar to the standard, single-fidelity BFGS quasi-Newton method, necessitating the constraint penalty function. Extension of the multi-fidelity concepts to constrained optimization is an area for future research. Figure 8 provides an XDSM diagram outlining the multi-fidelity optimization process. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Description** & **Equations** & **Factor** & **Offset** \\ \hline Cruise load factor & (21) & 25 & \(-\)0.001 \\ Cruise moment & (22) and (23) & 100 & \(-\)0.0005 \\ Maneuver load factor & (24) & 25 & \(-\)0.001 \\ KS stress at maneuver & (25) & 25 & \(-\)0.001 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Penalty factors and offsets for the different inequality constraints. A goal of this approach is to improve the scalability of multi-fidelity optimization with respect to the number of design variables. Popular methods such as Trust Region Model Management (TRMM) [15; 16; 17] and efficient global optimization [38] rely on building accurate surrogate models (or surrogate corrections to a lower-fidelity analysis) over as large a region as possible. Consequently, these approaches suffer from the curse of dimensionality inherent to the underlying surrogate methods. In contrast, the MF-BFGS method leverages the concept of an expected point (the result of a bound-constrained quadratic sub-problem, Step 2) to construct a localized, two-point correction (Step 4) focused along the search direction. MF-BFGS leverages high-fidelity gradient information (Step 8) to construct an approximate Hessian (Step 1) and find an objective-reducing search direction. While the method uses high-fidelity gradients and search directions, the line search (Step 5) implements corrected low-fidelity data (Step 6) with the goal of reducing overall cost. Another advantage of approximating the Hessian with high-fidelity gradients is that if multi-fidelity optimization fails to make progress, the process can revert to high-fidelity optimization (Step 10) with no loss of data. The numerical implementation of MF-BFGS is a Python wrapper around a slight modification of Nocedal's Limited-memory, Bound-constrained BFGS software (L-BFGS-B) [39; 40], compiled using the F2 Py Fortran-to-Python compiler. The modifications were required to break the iteration and manage the use of high- and multi-fidelity data within the optimization, as well as to manipulate the design data in the case of rejected steps. For a single fidelity of analysis, the method reduces to the original L-BFGS-B implementation.