Optimal control problems including partial differential equation (PDE) as well as integer constraints merge the combinatorial difficulties of integer programming and the challenges related to large-scale systems resulting from discretized PDEs. So far, the Branch-and-Bound framework has been the most common solution strategy for such problems. In order to provide an alternative solution approach, especially in a large-scale context, this article investigates penalization techniques. Taking inspiration from a well-known family of existing exact penalty algorithms, a novel improved penalty algorithm is derived, whose key ingredients are a basin hopping strategy and an interior point method, both of which are specialized for the problem class. A thorough numerical investigation is carried out for a standard stationary test problem. Extensions to a convection-diffusion as well as a nonlinear test problem finally demonstrate the versatility of the approach.
翻译:最佳控制问题,包括部分差异方程(PDE)以及整数限制,将整数编程的组合困难和与大型系统有关的难题合并起来,这些难题都来自分散的PDEs。到目前为止,分支和组合框架一直是解决这些问题的最常见解决办法。为了提供替代解决办法,特别是在大规模情况下,本条调查了惩罚性方法。在从众所周知的大家庭中汲取了现有精确惩罚算法的灵感之后,得出了一种新的改进惩罚算法,其关键成分是流域选择策略和内部点方法,这两种方法都专门针对问题类别。对标准的固定测试问题进行了彻底的数字调查。对吞并的扩展以及非线性测试问题最终显示了该方法的多功能性。