# Multi-Fidelity Gradient-Based Optimization for High-Dimensional Aeroelastic Configurations
## 2 Modeling and Analysis
### Multi-Fidelity, Gradient-Based Design Optimization
The algorithm minimizes a quadratic approximate model at the \(k^{\text{th}}\) iterate, \(m^{(k)}(\mathbf{D})\)[39],  
\[m^{(k)}(\mathbf{D})=f_{h}\Big{(}\mathbf{D}^{(k)}\Big{)}+\
abla f_{h}^{(k) \text{T}}\Big{(}\mathbf{D}-\mathbf{D}^{(k)}\Big{)}+\frac{1}{2}\Big{(}\mathbf{ D}-\mathbf{D}^{(k)}\Big{)}^{\text{T}}\mathbf{H}^{(k)}\Big{(}\mathbf{D}- \mathbf{D}^{(k)}\Big{)}, \tag{27}\]  
Figure 8: Extended design structure matrix diagram of the multi-fidelity BFGS process. Blue elements represent iterative processes (i.e., optimization convergence or line searching); red represents the fidelity switching decision; green represents the calculation of values involved in the optimization process. where the objective function and its gradient, \(f_{h}\) and \(\
abla f_{h}\), are evaluated using the high-fidelity analysis, and the positive definite approximate Hessian \(\mathbf{H}\) is approximated via the BFGS update over a series of steps, as described in [39; 40] (omitted here for conciseness). With the search direction defined by the the step minimizing this sub-problem, subject to the side constraints on the design variables, a line search may be performed using either the high-fidelity function or a multi-fidelity correction to the low-fidelity function. The optimization begins with the multi-fidelity line search and reverts to high-fidelity once the gradient norm is reduced to a user-selected level or the multi-fidelity trust region becomes too small to make progress, also defined by the user. In this work, the transition criteria are a gradient norm of \(1\times 10^{-3}\) or a trust region smaller than 0.001 (for normalized variables) for three successive iterations, which is aggressive in attempting to find a small trust region where the multi-fidelity model is sufficiently accurate rather than resorting to high-fidelity optimization. The reader is reminded here that the gradient and approximate Hessian always use the high-fidelity function; thus, the transition to single-fidelity optimization occurs as if the high-fidelity function was used from the start. While any method that enforces first-order consistency (i.e., function value and gradients) at the current design point may be used to construct the low-fidelity correction function, this and prior aero-structural design applications [41] implement the gradient-enhanced, Multi-Fidelity Polynomial Chaos (MFPC) implementation presented in [13; 42]. While polynomial chaos itself suffers from the curse of dimensionality, similar to other surrogate modeling techniques, here, the polynomials are limited to first order to ameliorate cost growth. Using both the high- and low-fidelity function and gradient information at two points (the selection of points will be explained momentarily), first-order additive and multiplicative corrective terms are computed in the form  
\[f_{h}(\mathbf{D})-f_{l}(\mathbf{D})=\hat{\alpha}(\mathbf{D})f_{l}(\mathbf{D})+ \hat{\delta}(\mathbf{D}). \tag{28}\]  
The linear polynomials \(\hat{\alpha}\) and \(\hat{\delta}\) are estimated, while \(f_{l}\) is the true low-fidelity computation. In typical MFPC usage, the high-fidelity function itself would be approximated, \(\tilde{f}_{h}\), but here, only the difference between the high- and low-fidelity functions is approximated. To ensure convergence of the multi-fidelity optimizer [14], one endpoint of the model is always the current design point. The other endpoint is the point nearest the expected point (in Euclidean distance) selected from the data previously computed over the course of optimization. It should be noted here that inside the optimization, the design variables are normalized to \(0\leq\mathbf{D}_{i}\leq 1\). Limiting the corrective functions to first-order polynomials, the low-fidelity model can be made first-order, consistent with the high-fidelity model at both points. During the first iteration, when there is no prior data, the high-fidelity function is also computed at the expected point to form the multi-fidelity approximation.