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

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, and solved by an outer approximation algorithm. However, both the relaxation and the algorithm are independent of any fixed discretization and can thus be formulated in function space. In a numerical evaluation for the case of a bounded number of switchings, we show that our approach does not only improve the dual bounds given by the straightforward continuous relaxation, but that our relaxation can even be solved more efficiently than the latter. In other words, our approach profits from the binarity of the controls, rather than suffering from it.

Parabolic optimal control problems with combinatorial switching constraints -- Part II: Outer approximation algorithm

“…In our companion paper [4], we propose an entirely different approach for the solution of (P). This approach is based on a tailored convexification of (P), which is built by cutting planes derived from finite-dimensional projections.…”

“…This approach is based on a tailored convexification of (P), which is built by cutting planes derived from finite-dimensional projections. The numerical experiments in [4] demonstrate that our convexification generally provides better dual bounds than the naive relaxation, which is obtained by replacing the binarity constraints u ∈ {0, 1} n by u ∈ [0, 1] n in the definition of D. Even more, the prototypical example in [4,Counterexample 3.1] shows that the naive relaxation does not give the closure of the convex hull of D in any L p (0, T ; R n ) in general. In addition, the naive relaxation may not benefit from the particular problem structure, as the the investigation of the min-up/min-down polytope in [29] demonstrates.…”

“…By contrast, our approach provably generates the closure of the convex hull in the limit. To be more precise, it is shown in [4] that conv(D) = k∈N V k , with the closure taken in L p (0, T ; R n ), see Theorem 2.2 below. Herein, the sets V k correspond to sets in R M k that, for prominent examples, can be shown to be polytopes for which the separation problem is tractable; see [4,Section 3.1 and 3.2].…”

“…To be more precise, it is shown in [4] that conv(D) = k∈N V k , with the closure taken in L p (0, T ; R n ), see Theorem 2.2 below. Herein, the sets V k correspond to sets in R M k that, for prominent examples, can be shown to be polytopes for which the separation problem is tractable; see [4,Section 3.1 and 3.2]. In this paper, we will combine the separation algorithm for V k with the passage to the limit k → ∞ in order to obtain an outer approximation algorithm whose iterates converge strongly in L 2 (0, T ; R n ) to the global minimizer of (P) with D replaced by conv(D).…”

“…The plan of the paper reads as follows: In Section 2, we state the standing assumptions on D and recall the main results of our companion paper [4] that will be needed for the construction and the analysis of our outer approximation algorithm. Section 3 is devoted to the convergence analysis of the outer approximation algorithm for k → ∞.…”