We present a general numerical solution method for control problems with state variables defined by a linear PDE over a finite set of binary or continuous control variables. We show empirically that a naive approach that applies a numerical discretization scheme to the PDEs to derive constraints for a mixed-integer linear program (MILP) leads to systems that are too large to be solved with state-of-the-art solvers for MILPs, especially if we desire an accurate approximation of the state variables. Our framework comprises two techniques to mitigate the rise of computation times with increasing discretization level: First, the linear system is solved for a basis of the control space in a preprocessing step. Second, certain constraints are just imposed on demand via the IBM ILOG CPLEX feature of a lazy constraint callback. These techniques are compared with an approach where the relations obtained by the discretization of the continuous constraints are directly included in the MILP. We demonstrate our approach on two examples: modeling of the spread of wildfire and the mitigation of water contamination. In both examples the computational results demonstrate that the solution time is significantly reduced by our methods. In particular, the dependence of the computation time on the size of the spatial discretization of the PDE is significantly reduced.

我们提出了一种控制问题的通用数值解方法，该方法具有状态变量，该状态变量由有限的二进制或连续控制变量集上的线性PDE定义。我们凭经验表明，将数值离散化方案应用于PDE以获得混合整数线性程序（MILP）的约束的幼稚方法会导致系统过大而无法使用MILP的最新求解器进行求解，尤其是当我们需要状态变量的精确近似值时。我们的框架包括两种技术，可减少离散化程度增加带来的计算时间的增加：首先，在预处理步骤中，将线性系统作为控制空间的基础进行求解。其次，某些约束只是通过惰性约束回调的IBM ILOG CPLEX功能按需施加的。将这些技术与通过连续约束离散化获得的关系直接包含在MILP中的方法进行了比较。我们通过两个示例展示了我们的方法：野火蔓延的建模和水污染的缓解。在两个示例中，计算结果均表明，我们的方法大大缩短了求解时间。尤其是，大大减少了计算时间对PDE空间离散化大小的依赖性。在两个示例中，计算结果均表明，我们的方法大大缩短了求解时间。尤其是，大大减少了计算时间对PDE空间离散化大小的依赖性。在两个示例中，计算结果均表明，我们的方法大大缩短了求解时间。尤其是，大大减少了计算时间对PDE空间离散化大小的依赖性。