发表机构
TU Dortmund University（多特蒙德工业大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出用McCormick松弛计算非凸PDE约束最优控制问题的近似对偶界，通过局部平均双线性项、推广理论至多维椭圆PDE、证明状态变量有效界并热启动OBBT的LP序列，实现更细离散化的高效数值求解。
AI 中文摘要
我们感兴趣的是使用McCormick不等式计算非凸最优控制问题的近似对偶界。为了处理与这些不等式收紧相关的数值负担，我们在PDE约束中对双线性项进行局部平均。我们将该领域的最新理论扩展到涉及双线性项的多维域上的椭圆型PDE。作为额外结果，我们还推广了必要假设，以允许在底层控制问题的总变差惩罚项的离散化选择上具有更大的灵活性。此外，我们证明了状态变量上有效界的存在性，这对于启用基于优化的界收紧（OBBT）过程来收紧McCormick不等式至关重要。由于状态变量上的理论界相当保守，OBBT对于获得尖锐的近似对偶界既至关重要，又是主要的计算瓶颈。我们展示了OBBT过程中产生的LP序列如何可以合理地热启动，从而使我们能够成功地将数值实验扩展到比以前更细的离散化。
英文摘要
We are interested in computing approximate dual bounds for nonconvex optimal control problems using McCormick inequalities. To handle the numerical burden concerned with the tightening of these inequalities, we locally average the bilinear term in the PDE constraint. We extend recent theory in this area to elliptic PDEs on multidimensional domains that involve bilinear terms. As an additional result, we also generalize the necessary assumptions to allow for more flexibility in the choice of the discretization for the total variation penalty term from the underlying control problem. Furthermore, we prove the existence of valid bounds on the state variable, which are crucial to enable the optimization-based bound-tightening (OBBT) procedure to tighten the McCormick inequalities. Since theoretical bounds on the state variable are rather conservative, OBBT is both crucial and the main computational bottleneck for deriving a sharp approximate dual bound. We show how the sequence of LPs arising during OBBT may sensibly be warmstarted, allowing us to successfully scale the numerical experiments to finer discretizations than what was possible before.
