Bilevel Optimal Control: Theory, Algorithms, and Applications

双层最优控制：理论、算法和应用

• 批准号: 313963978
• 负责人: Professor Dr. Stephan Dempe
• 金额: --
• 依托单位: Arbeitsgruppe Nichtlineare Optimierung
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2016
• 资助国家: 德国
• 起止时间: 2015-12-31 至 2022-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/313963978?language=en
• 关键词: Bilevel; Optimal; Control; Theory; Algorithms

Hierarchical optimization problems with two decision levels where at least one decision maker has to solve an optimal control problem are referred to as bilevel optimal control problems (BOCPs). Models of this structure typically arise from real-world applications which are related to e.g. pricing in energy markets, parameter estimation in process control, or data compression. BOCPs are inherently nonsmooth, infinite-dimensional programs with implicit constraints that suffer from inherent irregularity. This makes this problem class rather challenging.In this follow-up project, we plan to deepen our analysis on BOCPs of partial differential equations with a focus on potential optimality conditions and solution algorithms. Therefore, we are going to exploit two different single-level surrogates of the hierarchical model: The optimal-value-transformation which exploits the optimal value function of the lower level parametric optimization problem and the Karush-Kuhn-Tucker- (KKT-) transformation which replaces the lower level problem by first-order optimality conditions. Using appropriate upper estimates of the optimal value function which are available whenever this function is convex or concave, algorithms which iteratively refine the feasible set of relaxed surrogate problems of the optimal-value-transformation are imaginable. In the SPP's first stage, we derived a solution method of this type in the case of fully convex data. Here, the resulting convexity of the optimal value function was essential. Now, we want to investigate the situation where the optimal value function is concave which is natural in the context of parameter reconstruction. Furthermore, we are going to perform some numerical analysis in order to show that the solutions computed by our algorithms on the discretized stage converge to solutions in the function space setting under appropriate assumptions.Noting that the KKT-transformation of a BOCP with lower level inequality constraints is a complementarity-constrained optimization problem (MPCC) in function spaces, the latter problem class will be investigated carefully. We plan to derive new problem-tailored stationarity notions based on pointwise characterizations as well as associated constraint qualifications. Furthermore, second-order sufficient optimality conditions for the latter problem class will be derived. We aim for the construction of an active set method for the numerical solution of MPCCs in finite and infinite dimensions. Finally, we want to apply all our findings to prototypical applications from bilevel programming, namely parameter estimation problems as well as a problem of data compression using a bilevel measure approach. These models will be investigated from the viewpoint of optimality conditions and solution algorithms. We want to set up a corresponding collection of benchmark problems which allows a comparison of the derived theoretical results and numerical methods.

具有两个决策层的分层优化问题，其中至少一个决策者必须解决最优控制问题，称为双层最优控制问题（BOCPs）。这种结构的模型通常产生于现实世界的应用，例如，能源市场中的定价，过程控制中的参数估计，或数据压缩。BOCP是固有的非光滑，无限维的程序与隐含的约束，遭受固有的不规则性。这使得这类问题相当具有挑战性。在这个后续项目中，我们计划深化我们对偏微分方程BOCP的分析，重点关注潜在的最优性条件和求解算法。因此，我们将利用分层模型的两个不同的单级代理：利用较低级别参数优化问题的最优值函数的最优值变换和用一阶最优性条件替换较低级别问题的Karush-Kuhn-Tucker-（KKT-）变换。使用适当的上限估计的最优值函数，这是可用的，每当这个函数是凸或凹的，算法迭代细化的最优值变换的松弛代理问题的可行集是可以想象的。在SPP的第一阶段，我们推导出了这种类型的解决方法的情况下，完全凸数据。在这里，最优值函数的凸性是必不可少的。现在，我们想研究最优值函数是凹的情况，这在参数重构的上下文中是很自然的。此外，我们将进行一些数值分析，以显示我们的算法计算的解决方案在离散化阶段收敛到解决方案在适当的假设下，在功能空间setting.Noting的KKT-转换的BOCP与较低级别的不等式约束是一个互补约束优化问题（MPCC）在功能空间中，后一类问题将仔细研究。我们计划获得新的问题定制的平稳性概念的基础上逐点表征以及相关的约束资格。此外，第二阶充分最优性条件后一类问题将被导出。我们的目标是建设一个积极的集方法的数值解MPCCs在有限和无限维。最后，我们希望将我们所有的研究结果应用到双层规划的原型应用中，即参数估计问题以及使用双层测量方法的数据压缩问题。这些模型将从最优性条件和求解算法的角度进行研究。我们希望建立一个相应的基准问题的集合，它允许导出的理论结果和数值方法的比较。