An improved penalty algorithm using model order reduction for MIPDECO problems with partial observations
针对含整数约束和部分观测的线性时变偏微分方程最优控制问题，提出一种结合内点法、预处理和模型降阶的改进惩罚算法，并通过热方程和对流扩散问题验证其有效性。
Abstract This work addresses optimal control problems governed by a linear time-dependent partial differential equation (PDE) as well as integer constraints on the control. Moreover, partial observations are assumed in the objective function. The resulting problem poses several numerical challenges due to the mixture of combinatorial aspects, induced by integer variables, and large scale linear algebra issues, arising from the PDE discretization. Since classical solution approaches such as the branch-and-bound framework are typically overwhelmed by such large-scale problems, this work extends an improved penalty algorithm proposed by the authors, to the time-dependent setting. The main contribution is a novel combination of an interior point method, preconditioning, and model order reduction yielding a tailored local optimization solver at the heart of the overall solution procedure. A thorough numerical investigation is carried out both for the heat equation as well as a convection-diffusion problem demonstrating the versatility of the approach.