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The combinatorial integral approximation (CIA) decomposition suggests solving mixed-integer optimal control problems 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 constructed feasible solution is guaranteed. 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 in the case of two controls. Finally, we present numerical experiments to highlight the proposed algorithm’s advantages in terms of run time and solution quality.

Tailored Mixed-Integer Optimal Control policies for real-world applications usually have to avoid very short successive changes of the active integer control. Minimum dwell time (MDT) constraints express this requirement and can be included into the combinatorial integral approximation decomposition, which solves mixed-integer optimal control problems (MIOCPs) to $$\epsilon $$ ϵ -optimality by solving one continuous nonlinear program and one mixed-integer linear program (MILP). Within this work, we analyze the integrality gap of MIOCPs under MDT constraints by providing tight upper bounds on the MILP subproblem. We suggest different rounding schemes for constructing MDT feasible control solutions, e.g., we propose a modification of Sum Up Rounding. A numerical study supplements the theoretical results and compares objective values of integer feasible and relaxed solutions.

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