  We present a numerical method to efficiently solve optimization problems
governed by large-scale nonlinear systems of equations, including discretized
partial differential equations, using projection-based reduced-order models
accelerated with hyperreduction (empirical quadrature) and embedded in a
trust-region framework that guarantees global convergence. The proposed
framework constructs a hyperreduced model on-the-fly during the solution of the
optimization problem, which completely avoids an offline training phase. This
ensures all snapshot information is collected along the optimization
trajectory, which avoids wasting samples in remote regions of the parameters
space that are never visited, and inherently avoids the curse of dimensionality
of sampling in a high-dimensional parameter space. At each iteration of the
proposed algorithm, a reduced basis and empirical quadrature weights are
constructed precisely to ensure the global convergence criteria of the
trust-region method are satisfied, ensuring global convergence to a local
minimum of the original (unreduced) problem. Numerical experiments are
performed on two fluid shape optimization problems to verify the global
convergence of the method and demonstrate its computational efficiency;
speedups over 18x (accounting for all computational cost, even cost that is
traditionally considered "offline" such as snapshot collection and data
compression) relative to standard optimization approaches that do not leverage
model reduction are shown.
