Optimal control problems (OCP) containing both integrality and partial differential
equation (PDE) constraints are very challenging in practice. The most wide-spread solution approach
is to first discretize the problem, resulting in huge nonlinear mixed-integer optimization problems
that can be solved to proven optimality only in very small dimensions. In this paper, we propose a
novel outer approximation approach to efficiently solve such OCPs in the case of certain semilinear
elliptic PDEs with static integer controls over arbitrary combinatorial structures. The basic idea is
to decompose the OCP into an integer linear programming (ILP) master problem and a subproblem
for calculating linear cutting planes. These cutting planes rely on the pointwise concavity of the
PDE solution operator in terms of the control variables, which we prove in the case of PDEs with a
non-decreasing convex nonlinear part. The decomposition allows to use standard solution techniques
for ILPs as well as for PDEs. We further benefit from reoptimization strategies due to the iterative
structure of the algorithm. Experimental results show that the new approach is capable of solving
the combinatorial OCP of a semilinear Poisson equation with up to 230 binary controls to global
optimality within a 5h time limit. Applied to the screened Poisson equation, problems with even
2200 binary controls are globally solvable