Mixed Integer PDE Constrained Optimization for the Control of a Wildfire Hazard

Abstract

We derive an optimization problem for a mission planning problem of a firefighting department challenged by a wildfire. Here the fire is modeled using partial differential equations (PDEs), and the response from the firefighters is modeled as a dynamic network flow. The firefighters influence the spread of the wildfire, and vice versa, the fire restricts the movement options of the firefighters. These mutual interactions have to be incorporated into the model. The presented approach to formulate this problem mathematically is to replace the infinite dimensional constraints imposed by the PDE by a finite dimensional system. These systems however tend to be very large even for a moderate resolution of the approximation. This causes a direct approach using a finite difference method to be outperformed by a new method, in which the PDE is solved in a pre-optimization step. We demonstrate the superiority of this approach in a computational study, where both methods are compared for various approximation resolutions.