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.

所述组合积分近似（CIA）分解表明通过求解一个连续的非线性控制问题和一个混合整数线性规划（MILP）求解混合整数最优控制问题。通过向MILP添加总变化量的约束，可以避免不切实际的频繁切换。在这项工作中，我们提出了一种快速的启发式方法来解决该CIA问题，并研究在什么情况下可以保证所构造的可行解决方案的最优性。在本文的第二部分中，我们给出了松弛连续控制轨迹与两个可行情况下的整数个可行轨迹之间的积分间隙的严格边界。最后，我们进行数值实验以突出提出的算法在运行时间和解决方案质量方面的优势。