Parabolic Optimal Control Problems with Combinatorial Switching Constraints, Part I: Convex Relaxations

.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. The convex relaxation is built by cutting planes derived from finite-dimensional projections, which can be studied by means of polyhedral combinatorics. A numerical example for the case of a bounded number of switchings shows that our approach can significantly improve the dual bounds given by the straightforward continuous relaxation, which is obtained by relaxing binarity constraints.KeywordsPDE-constrained optimizationswitching time optimizationconvex relaxationsMSC codes90C1190C57