Parabolic Optimal Control Problems with Combinatorial Switching Constraints, Part II: Outer Approximation Algorithm

SIAM Journal on Optimization, Volume 34, Issue 2, Page 1295-1315, June 2024.
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. In a companion paper [C. Buchheim, A. Grütering, and C. Meyer, SIAM J. Optim., arXiv:2203.07121, 2024], we describe the [math]-closure of the convex hull of feasible switching patterns as the intersection of convex sets derived from finite-dimensional projections. In this paper, the resulting outer description is used for the construction of an outer approximation algorithm in function space, whose iterates are proven to converge strongly in [math] to the global minimizer of the convexified optimal control problem. The linear-quadratic subproblems arising in each iteration of the outer approximation algorithm are solved by means of a semismooth Newton method. A numerical example in two spatial dimensions illustrates the efficiency of the overall algorithm.