  We present an augmented Lagrangian trust-region method to efficiently solve
constrained optimization problems governed by large-scale nonlinear systems
with application to partial differential equation-constrained optimization. At
each major augmented Lagrangian iteration, the expensive optimization
subproblem involving the full nonlinear system is replaced by an empirical
quadrature-based hyperreduced model constructed on-the-fly. To ensure
convergence of these inexact augmented Lagrangian subproblems, we develop a
bound-constrained trust-region method that allows for inexact gradient
evaluations, and specialize it to our specific setting that leverages
hyperreduced models. This approach circumvents a traditional training phase
because the models are built on-the-fly in accordance with the requirements of
the trust-region convergence theory. Two numerical experiments (constrained
aerodynamic shape design) demonstrate the convergence and efficiency of the
proposed work. A speedup of 12.7x (for all computational costs, even costs
traditionally considered "offline" such as snapshot collection and data
compression) relative to a standard optimization approach that does not
leverage model reduction is shown.
