Parabolic optimal control problems with combinatorial switching constraints -- Part I: Convex relaxations

TL;DR

This paper introduces a convex relaxation approach for optimal control problems with binary controls and combinatorial switching constraints, improving dual bounds over standard relaxations.

Contribution

It develops a novel convex relaxation based on polyhedral combinatorics to better handle combinatorial switching constraints in control problems.

Findings

Convex relaxation significantly improves dual bounds.

Polyhedral combinatorics effectively characterizes feasible switching patterns.

Numerical example demonstrates the approach's effectiveness.

Abstract

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.