An Improved Penalty Algorithm using Model Order Reduction for MIPDECO problems with partial observations

TL;DR

This paper presents an advanced solution method for complex optimal control problems involving PDEs with integer constraints and partial observations, combining penalty algorithms, model order reduction, and interior point methods to improve computational efficiency.

Contribution

The work introduces a novel combination of interior point methods, preconditioning, and model order reduction within an improved penalty algorithm for time-dependent PDE-constrained mixed-integer problems.

Findings

Effective handling of large-scale PDE-constrained MIP problems

Versatile approach demonstrated on Poisson and convection-diffusion problems

Significant computational improvements over classical methods

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 a Poisson problem as well as a convection-diffusion problem demonstrating the versatility of the approach.