论文标题
组合切换约束的抛物线最佳控制问题 - 第一部分：凸放松
Parabolic optimal control problems with combinatorial switching constraints -- Part I: Convex relaxations
论文作者
论文摘要
我们考虑了部分微分方程的最佳控制问题，其中控件采用二进制值但在时间范围内有所不同，因此可以将它们视为动态开关。开关模式可能会受到组合约束，例如，开关总数上的上限或两个开关之间的时间下限。尽管这种组合约束通常被视为在启发式后处理中处理的额外并发症，但我们方法的核心是研究所有可行的切换模式的凸壳，以定义控制问题的紧密凸放放松。凸松弛是通过切割有限维投影的切割平面来构建的，可以通过多面体组合学对其进行研究。一个有界数的开关数字的数字示例表明，我们的方法可以显着改善直接连续放松给出的双重界限，这是通过放松二进制约束而获得的。
We consider optimal control problems for partial differential equations where the controls take binary values but vary over the time horizon, they can thus be seen as dynamic switches. The switching patterns may be subject to combinatorial constraints such as, e.g., an upper bound on the total number of switchings or a lower bound on the time between two switchings. While such combinatorial constraints are often seen as an additional complication that is treated in a heuristic postprocessing, the core of our approach is to investigate the convex hull of all feasible switching patterns in order to define a tight convex relaxation of the control problem. The convex relaxation is built by cutting planes derived from finite-dimensional projections, which can be studied by means of polyhedral combinatorics. A numerical example for the case of a bounded number of switchings shows that our approach can significantly improve the dual bounds given by the straightforward continuous relaxation, which is obtained by relaxing binarity constraints.
