An improved penalty algorithm using model order reduction for MIPDECO problems with partial observations

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.

Special issue for SIMAI 2020–2021: large-scale optimization and applications

“…In [9] the authors deal with the numerical solution of mixed integer PDE constrained optimization (MIPDECO) problem with a linear time-dependent constraint and integer constraints on the control. An improved penalty algorithm (IPA), recently developed by the authors in [10], is suitably adapted to this linear timedependent setting, having partial observations in the objective function.…”