The combinatorial integral approximation (CIA) decomposition suggests to solve mixed-integer optimal control problems (MIOCPs) by solving one continuous nonlinear control problem and one mixed-integer linear program (MILP). Unrealistic frequent switching can be avoided by adding a constraint on the total variation to the MILP. Within this work, we present a fast heuristic way to solve this CIA problem and investigate in which situations optimality of the provided solution is guaranteed. Our proof reveals a link from the CIA problem to scheduling theory. In the second part of this article, we show tight bounds on the integrality gap between a relaxed continuous control trajectory and an integer feasible one. Finally, we present numerical experiments to highlight the advantages of the proposed algorithm in terms of run time and solution quality.
Sager, Sebastian and Zeile, Clemens, "On Mixed-Integer Optimal Control with Constrained Total Variation of the Integer Control", MathOpt Group, Faculty of Mathematics, Otto-von-Guericke-University Magdeburg, Germany.