An Augmented Lagrangian Trust‐Region Method With Inexact Gradient Evaluations to Accelerate Constrained Optimization Problems Using Model Hyperreduction

Abstract

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 (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.