Efficient Solution of Discrete Subproblems Arising in Integer Optimal Control with Total Variation Regularization

We consider a class of integer linear programs (IPs) that arise as discretizations of trust-region subproblems of a trust-region algorithm for the solution of control problems, where the control input is an integer-valued function on a one-dimensional domain and is regularized with a total variation term in the objective, which may be interpreted as a penalization of switching costs between different control modes. We prove that solving an instance of the considered problem class is equivalent to solving a resource constrained shortest path problem (RCSPP) on a layered directed acyclic graph. This structural finding yields an algorithmic solution approach based on topological sorting and corresponding run time complexities that are quadratic in the number of discretization intervals of the underlying control problem, the main quantifier for the size of a problem instance. We also consider the solution of the RCSPP with an $A^*$ algorithm. Specifically, the analysis of a Lagrangian relaxation yields a consistent heuristic function for the $A^*$ algorithm and a preprocessing procedure, which can be employed to accelerate the $A^*$ algorithm for the RCSPP without losing optimality of the computed solution. We generate IP instances by executing the trust-region algorithm on several integer optimal control problems. The numerical results show that the accelerated $A^*$ algorithm and topological sorting outperform a general purpose IP solver significantly. Moreover, the accelerated $A^*$ algorithm is able to outperform topological sorting for larger problem instances.