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.

翻译：我们提出了一种增广拉朗日信赖域方法，用于高效求解由大规模非线性系统控制的约束优化问题，并应用于偏微分方程约束优化。在每一次主要的增广拉格朗日迭代中，涉及完整非线性系统的昂贵优化子问题被一个基于经验求积的超约简模型所取代，该模型是即时构建的。为确保这些不精确增广拉格朗日子问题的收敛性，我们开发了一种允许不精确梯度评估的边界约束信赖域方法，并将其专门应用于我们利用超约简模型的具体场景。这种方法绕过了传统的训练阶段，因为模型是根据信赖域收敛理论的要求即时构建的。两个数值实验（约束空气动力学形状设计）证明了所提方法的收敛性和效率。相对于不利用模型降阶的标准优化方法，该方法实现了12.7倍的加速（针对所有计算成本，甚至包括传统上被视为"离线"的成本，如快照收集和数据压缩）。