By Alexandra Gruetering. Doctorante de la Universidad Técnica de Dortmund, Alemania.
Seminar Date: 2023-02-16
We consider optimal control problems for partial differential equations where the controls take binary values but vary over the time horizon, they can thus be seen as dynamic switches. The switching patterns may be subject to combinatorial constraints such as, e.g., an upper bound on the total number of switchings or a lower bound on the time between two switchings. While such combinatorial constraints are often seen as an additional complication that is treated in a heuristic postprocessing, the core of our approach is to investigate the convex hull of all feasible switching patterns in order to define a tight convex relaxation of the control problem. Solving the resulting convex relaxations by an outer approximation approach and embedding this into a tailored branch-and-bound scheme, our objective is to obtain globally optimal solutions. In this talk, we will discuss the overall branch-and-bound algorithm, which intends to implicitly determine the switching structure of the optimal control without preceding discretization. We also show that the convex relaxation can be built by cutting planes derived from finite-dimensional projections, which can be studied by means of polyhedral combinatorics.